Receive him then in the Lord with all joy: prosdechesthe (2PPMM) oun auton en kurio meta pases charas: (Mt 10:40,41; Lk 9:5; Jn 13:20; Ro 16:2; 1Cor 16:10; 2Cor 7:2; Col 4:10; 3Jn 1:10) (Isa 52:7; Lk 2:10,11; Acts 2:46; 8:8; Ro 10:15; Eph 4:9, 10, 11, 12)

Receive (4327) (prosdechomai [word study]  from pros = in compound Greek words implies motion or direction toward + dechomai [word study] = a deliberate and ready reception) means to accept favorably, to receive one into intercourse/companionship, to give access to oneself or receive to oneself. 

Paul commands the Philippian saints to continually (present imperative) be earnestly expecting, looking forward to and favorably accepting Epaphroditus. The middle voice (reflexive) indicates they were to receive him to themselves (reflexive) and the prefixed preposition (pros) in its root meaning signifies “facing” which implies fellowship.

and hold men like him in high regard: kai tous toioutous entimous echete, (2PPAM): (2Cor 10:18; 1Thes 5:12; Heb 13:17) (Acts 28:10; 1Cor 16:18; 1Ti 5:17)

Hold (2192) (echo) means to have or hold and is in the present imperative commanding them to continually value Epaphroditus highly.

because he came close to death for the work of Christ:hoti dia to ergon Christou mechri thanatou eggisen, (3SAAI): (1Cor 15:53; 16:10)

Came close (1448) (eggizo [word study] from eggús = near) means to move nearer to a reference point, to be at hand or nigh.

Death (2288) (thanatos from thnesko = to die) refers to physical death, of separation of the soul from the body.

Work (2041) (ergon from ergo = to work) refers to toil as an effort or occupation.

See related topic - click for discussion of what constitutes a "good deed" in the sight of God.

risking his life to complete what was deficient  in your service to me: paraboleusamenos (AMPMSN) te psuche hina anaplerose (3SAAS) to humon husteremates pros me leitourgias: (Phil 2:17,27; 1:19,20; Mt 25:36, 37, 38, 39, 40; Acts 20:24; Ro 16:4; 2Cor 12:15; Rev 12:11) (Phil 4:10,18; 1Cor 16:17; Philemon 1:13)

“having gambled with his life” (LIghtfoot)

Risking his life (3850) (paraboleuomai from parabolos = venturesome, reckless) literally means to throw aside (para  = aside + ballo = to throw), hence to expose to danger.

In the present context it means Epaphroditus was willing to die. In those days when you visited prisoners held by the Romans, the visitor was often prejudged as a criminal also. Therefore the visitor exposed himself to danger just by being near those who were considered dangerous. This word came to also be used in the sense of playing the gambler or playing dice because high sums were often at stake. In a sense Epaphroditus was gambling with his life for the sake of God's kingdom. The word was used in the papyri of one who in the interest of friendship had exposed himself to dangers as an advocate in legal strife by taking his clients' cause even up to emperors.

Paraboleuomai was later used of merchants who for the sake of gain exposed themselves to death. The word was used of a fighter in the arena who exposed himself to the dangers of the arena. In the post-apostolic church there were societies of men and women who called themselves "the paraboloni" or "the riskers or gamblers". The risked their lives by ministering the sick and imprisoned and they saw to it if possible that martyrs and sometimes even their enemies would receive an honorable burial. In Carthage during the great plague of 252AD, Cyprian, the bishop, showed remarkable courage, taking on himself the care of the sick and  urging of his flock to nurse them and bury those who died from the pestilence. Cyprian's conduct like a light in the darkness contrasted to the practice of the pagans who threw the corpses out of the plague-infested city and actually ran from them in terror! Such is the transforming effect of the gospel.

Barclay adds that paraboleuomai was

a gambler’s word and means to stake everything on a turn of the dice. Paul is saying that for the sake of Jesus Christ Epaphroditus gambled his life.

Life (590) (psuche from psucho = to breathe, blow) refers to whole person particularly inner, immortal person who lives in the mortal body. It describes the breath of life as the vital force which animates the body and shows itself in breathing. Dichotomists view man as consisting of two parts material and immaterial, with spirit and soul denoting the immaterial and bearing only a functional and not a metaphysical difference. Trichotomists also view man as consisting of two parts, but with spirit and soul representing in some contexts a real subdivision of the immaterial. In the latter view psuche contrasts with soma =  body, and pneúma = spirit (see 1Th 5:23).

To complete (378) (anapleroo from aná = up or as an emphatic + pleroo = to fill) means to fill up or supply a deficiency.

Deficient  (5303) (husterema from hustereo = to lack, fall behind) means a deficit as that which is behind or that which is lacking)

Service  (3009) (leitourgia from leitourgeo = to be a public servant, to perform religious or charitable function, to minister) generally used of a servant of a superior and suggests a function to be discharged or a necessary service to be rendered. 

Leitourgia is the word Paul used to describe himself as...

being poured out as a drink offering upon the sacrifice and service of your faith (see note Philippians 2:17)