The Catechism of the Catholic Church has this to say on satisfaction:
1459 Many sins wrong our neighbor. One must do what is possible in order to repair the harm (e.g., return stolen goods, restore the reputation of someone slandered, pay compensation for injuries). Simple justice requires as much. But sin also injures and weakens the sinner himself, as well as his relationships with God and neighbor. Absolution takes away sin, but it does not remedy all the disorders sin has caused.62 Raised up from sin, the sinner must still recover his full spiritual health by doing something more to make amends for the sin: he must "make satisfaction for" or "expiate" his sins. This satisfaction is also called "penance."
1460 The penance the confessor imposes must take into account the penitent's personal situation and must seek his spiritual good. It must correspond as far as possible with the gravity and nature of the sins committed. It can consist of prayer, an offering, works of mercy, service of neighbor, voluntary self-denial, sacrifices, and above all the patient acceptance of the cross we must bear. Such penances help configure us to Christ, who alone expiated our sins once for all. They allow us to become co-heirs with the risen Christ, "provided we suffer with him."63
The satisfaction that we make for our sins, however, is not so much ours as though it were not done through Jesus Christ. We who can do nothing ourselves, as if just by ourselves, can do all things with the cooperation of "him who strengthens" us. Thus man has nothing of which to boast, but all our boasting is in Christ ... in whom we make satisfaction by bringing forth "fruits that befit repentance." These fruits have their efficacy from him, by him they are offered to the Father, and through him they are accepted by the Father.64
62 DS 1712 (Council of Trent, 1551)
63 ⇒ Rom 8:17; ⇒ Rom 3:25; ⇒ 1 Jn 2:1-2; cf. DS 1690 (Council of Trent, 1551)
64 DS 1691 (Council of Trent, 1551); cf. ⇒ Phil 4:13; ⇒ 1 Cor 1:31; ⇒ 2 Cor 10:17; ⇒ Gal 6:14; ⇒ Lk 3:8.
The Biblical references are [all RSV-CE]:
Rom 8:17 — and if children, then heirs, heirs of God and fellow heirs with Christ, provided we suffer with him in order that we may also be glorified with him.
Rom 3:25 — ... Christ Jesus, whom God put forward as an expiation by his blood, to be received by faith. This was to show God's righteousness, because in his divine forbearance he had passed over former sins.
1 Jn 2:1–2 — If anyone does sin, we have an advocate with the Father, Jesus Christ the righteous; and he is the expiation for our sins, and not for ours only but also for the sins of the whole world.
Phil 4:13 — I can do all things in him who strengthens me.
1 Cor 1:31 — Let him who boasts boast in the Lord.
2 Cor 10:17 — Let him who boasts boast of the Lord.
Gal 6:14 — But far be it from me to glory except in the cross of our Lord Jesus Christ, by which the world has been crucified to me, and I to the world.
Lk 3:8 — Bear fruits that befit repentance, and do not begin to say to yourselves, 'We have Abraham as our father'; for I tell you, God is able from these stones to raise up children to Abraham.
The biblical basis for penance is thus Romans 8:17, the patient acceptance of the cross we must bear to be conformed to Christ who bore his cross for the sin of the world. The penance imposed by the priest is a small suffering in our own life to remind us of Jesus' ultimate sacrifice in his; because of his suffering, he is able to support us in ours. And so, having accepted our own little cross, we might bear the fruits of repentance: the "turning back to God" and the change of life that must engender.