Convinced of this: kai touto pepoithos (RAPMSN):  (2:24; Ac20:25)

"And I am confidently certain of this." (Barclay) 
"And a voice within assures me, that so it will be" (Lightfoot 1)|
"And having come to this settled conviction (namely, that to remain in the flesh is more needful for you)" (Wuest)
"That is why (it is probably more necessary for you that I should stay here on earth) I feel pretty well convinced" (Phillips)

Convinced (peitho) is used 6x in this short epistle (1:6, 1:14, 1:25, 2:24, 3:3 3:4) and speaks of Paul's settled conviction which is the result of a past completed process of turning this matter over in his mind until he was persuaded of it. "Convinced" is in the perfect tense implying that Paul has been persuaded at some point in time in the past and was presently thoroughly convinced with this settled conviction.

As Barclay says Paul was

"confidently certain". 

I know that I will remain and continue with you all for your progress and joy in the faith: oida (1SRAI) hoti meno (1SFAI) kai parameno (1SFAI) pasin humin eis ten humon prokopenkai charan tes pisteos:  (Lu22:32; Jn21:15-17; Ac11:23; 14:22; Ro1:11 1:12; 15:18 15:29; 2Co1:24;  Eph4:11, 12, 13) (Ro5:2; 15:13; 1Pe1:8) (See Torrey's Topic on "Joy") 

"that I will remain, and I will be with you and beside you all to help you along the road, and to increase the joy of your faith" (Barclay) 
"I shall continue here and abide with you all; that I may promote your advance in the faith and your joy in believing." (Lightfoot)
"I know that I shall remain and continue alive with you all for your progress and joy in your faith" (Wuest)
"that I shall not leave this world yet, but shall be able to stand by you, to help you forward in Christian living and to find increasing joy in your faith" (Phillips)

Continue (parameno) is literally to remain near or continue near as Barclay brings out in his translation "I will be with you and beside you". The preposition "para" means beside which stresses the place of Paul's abiding. Right by their side. 

 Barclay adds that parameno

"means to wait beside a person ever ready to help."

Rienecker says that in koine Greek parameno conveyed the idea "to remain in service" or to "remain at someone else's disposal" which helps us understand the encouragement this truth must have conveyed to the saints at Philippi.

Wuest comments that

"Paul had turned over in his mind the need which the Philippian saints had of his ministry, and had come to the settled conviction that they needed him more than he needed to go to heaven. That was just like Paul. He lived a crucified life, dead to self, ever setting even his legitimate desires aside in order that he might serve others. And so he tells them that he will remain on earth with them. While Paul had no active choice in the matter, yet he believed that the servant of the Lord is immortal until his work is done. Thus, if the Philippians needed his ministry, that fact would indicate that he was not to die at that time by the hand of Rome, but that he would be released and thus be able to minister to the spiritual needs of the saints."

"Progress" (Proskope) is from same word in v12, referring to the pioneer advance of the gospel there, and here, to the Christian progress which the Philippians would make under the ministry of the apostle, a progress in new paths of Christian conduct and service which would otherwise not be possible. “Progress” pictures trail blazing so that an army can advance (see  v12). Paul wanted to cut a new path for the Philippians to follow to victory.

And in this verse we find this golden thread of joy that is interwoven throughout this epistle (Click for all 12v) The increasing of their faith would result in the increasing of their joy. Paul will continue abiding with the the Philippians saints in order to promote the progress of their faith and joy of their faith. Both words are connected by one definite article in Greek. It is the joy of the faith. Their progress will result in their joy. How about you dear reader? Is the Spirit bearing the fruit of joy in your walk of faith? If not, you might do a spiritual inventory to determine whether you are making "progress". There is no neutrality in the walk of faith. We are either going forward or going backward. Press on dear brother or sister. Do not lose heart. Lay aside every encumbrance and the sin which so easily entangles you and press on toward the goal for the only prize worth possessing.

"The faith" (pistis) is a phrase that occurs  some 38 times in the NASB and twice in this epistle (here and 1:27).

The faith can refer to personal saving faith but in both uses in this epistle (and ~ 50% of the 38 NT uses) "the faith" refers not to the ACT of believing but rather to WHAT is believed, especially the gospel (Click for more discussion of this concept in 2Timothy 4:7). 

"so that you may have still further grounds for boasting in Christ because of me, when once again I come to visit you." (Barclay)
"and that you on your part may have in me fresh cause for boasting in Christ, when you see me present among you once more." (Lightfoot)
"So you can look forward to making much of me as your minister in Christ when I come to see you again!" (Phillips),
"In order that your rejoicing may abound in Christ Jesus through me by reason of my personal presence with you again." (Wuest),
"so that through my being with you again your joy in Christ Jesus will overflow on account of me" (NIV),
"Then, when I visit you again, you will have good reason to take great pride in Christ Jesus because of me." (CEV),
"that your boasting may abound in Christ Jesus through me by my presence again with you." (Darby),
"that your rejoicing for me may be more abundant in Jesus Christ by my coming to you again" (NKJV)
So by coming to you again, I want to give you even more reason to have pride in Christ Jesus with me." (GWT)

"So that" marks a specific purpose, the giving to the Philippians saints of an abundant ground for boasting.

"Proud confidence" (glorying) (kauchema) is the result of boasting. This verse is somewhat difficult to understand in the NASB translation. The old ASB (1901) is easier to understand:

"that your glorying may abound in Christ Jesus in me through my presence with you again."

 A T Robertson emphasizes that

"In Christ Jesus” as the basis for the glorying" 

Vine writes that

"it looks as if the pronoun “your” points to their glorying in Paul. But in the original  (Greek) the case may be the objective, and the presence of the article with “glorying” indicates that the meaning is, “the glorying in you [that is, “his glorying in them”] may abound in me in Christ Jesus.” Either meaning is possible."

The idea seems to be that the saints in Philippi would glory, boast or exult because the one who had taught them about Christ would be with them again and that their "proud confidence" or exulting would ultimately be in Christ Jesus, the Source of true joy for all believers.

Thus the NIV translates it as

"your joy (literally "your boasting") in Christ Jesus will overflow on account of me".

The BBE offers a well worded compromise translation

"So that your pride in me may be increased in Christ Jesus through my being present with you again."

Expositor's comments that

"The emphasis is not on the action itself, but on the basis for it. As the Philippians would experience the progress and joy that Paul's labors among them would produce, they would have new and greater reasons for overflowing with joy. This reason for glorying (their "proud confidence") would be found "in Christ Jesus," Of course, but its immediate occasion would be "on account of me" (en emoi), said Paul. His ministry among them would enable them to see more clearly the riches of their salvation in Christ."

MacDonald adds that

"Through his being spared for longer life and service on earth, the Philippians would have added cause for rejoicing in the Lord when he would visit them once again. Can you not imagine how they would throw their arms around him and kiss him, and praise the Lord with great joy when he would arrive at Philippi?"

Barnes notes that

"Through the mercy and grace of Christ, if he (Paul) was spared, his deliverance would be traced to Christ, and they (saints in Philippi) would rejoice together in One (Christ Jesus) Who had so mercifully delivered him. Their joy would not only be that he was delivered, but that he was permitted to see them again."

"Abound" (perisseuo) be in excess, overflow and is a favorite of Paul in this epistle to the Philippians (Click here for all 4 uses in Php) The point is, as Paul lived on fruitfully, their joy and confidence would continue to overflow because of Christ’s working in him, not because of anything he himself did by his own ability. Paul wanted their proud confidence, glorying or exultation to continually overflow, to exceed a fixed number or measure, to exist in abundance and to be over and above. A synonym we might use today would be the "abundant" Christian life. Do you have a mentor like Paul who was gone for some time and now has returned with the result that you are now overflowing with exultation to Christ Jesus?

"In Christ Jesus" as Wuest notes indicates that

"Christ Jesus is the Sphere in which these blessings are enjoyed, the Sphere in the sense that He made them possible through the blood of His Cross, and in the sense that He is the joy of the believer’s life, the One who completely satisfies. Paul is the human instrument through whom God works to bring these joys to the Philippians by means of his personal presence with them again." Christ Jesus ultimately is the only One in which we should boast (1Cor1:31)

The boasting of the Philippian saints did not center on Paul, but had Christ as the Object, for He was the One Whose Person, righteousness, and salvation were continually emphasized by Paul for their focused attention and their comfort. These saints would rejoice with overflowing if and when Paul returned to them but even then their rejoicing as has been repeatedly emphasized would ultimately be in Christ Jesus.

As Matthew Henry reminds us

"All our joys should terminate in Christ. Our joy in good ministers should be our joy in Christ Jesus for them, for they are but the friends of the bridegroom, and are to be received in His name, and for His sake."

“Coming” (3952) (parousia from para = with +  ousia =being = the participial form of the verb eimi = to be) (Click in depth word study on parousia) literally means a "being beside" or a presence. Parousia in the NAS is translated "coming" 22 times and as "presence" twice.

The 24 uses of parousia: Matthew 4x; 1 Corinthians 2x; 2 Corinthians 3x; Philippians 2x; 1 Thessalonians 4x; 2 Thessalonians 3x;