This position is not universally accepted. Some Wesleyan theologians and Eastern Orthodox strongly oppose this. Isa 53, however, is very clear on this subject.
Here are the points one must touch when looking at Isa 53.
- the Servant was wounded for our transgressions
- the Servant was inflicted with the stripes of whips for our healing
- the Servant had the iniquities of us all laid upon him
- the Servant was but put to death for the transgression of the people "to whom the stroke was due,"
- Yahweh was pleased to bruise the Servant
- Yahweh put the Servant to grief
- the Servant's soul was offered as a sin offering
- Yahweh will see the travail of the Servant's soul and be satisfied
- the Servant will bear their iniquities
- the Servant bore the sins of many
In sum, God wounded, bruised, whipped, grieved, and put Jesus to death as a sin offering for our transgressions, and thereby was "satisfied." If this isn't penal satisfaction, then I don't know how else the prophet or the Apostles could make it clearer.