The Sabbath is so significant that Jesus' ministry formally began on a Sabbath and ended on a preparation day just before another Sabbath (John 19:31)! We see Him open His ministry in Luke 4:16-19, where He gives His mission statement. By quoting Isaiah 61:1-2 in His inaugural sermon, Jesus identifies His mission as setting people free from bondage. He specifically mentions freeing the poor (weak, without power), brokenhearted, captive, blind, and oppressed.
"The acceptable year of the LORD" is not when God is acceptable to us, but when God, in His sovereign mercy, moves to make us acceptable to Him. It is a time when He chooses to deliver people. More specifically, it refers to two Old Testament institutions, either the seventh year land Sabbath or the Jubilee year. Israelites considered these years liberators of the oppressed. During them, the land lay fallow and what food it produced on its own went to the poor, dispossessed, and animals. Slaves were freed and debts remitted. During Jubilee years, debtors received back their land lost due to mismanagement.
Jesus says in verse 21, "Today this Scripture is fulfilled in your hearing." It was a Sabbath, and through the typology, Christ is clearly showing that His redemptive mission included the liberating intent of the Sabbaths, weekly and annual. In Mark 2:27, Jesus says, "The Sabbath was made for man." God made it to equip us to come out of spiritual slavery—and even more so, to help us in staying out.