This is the beginning of Jesus' public ministry, we could call it His inaugural address. Jesus began His ministry on a Sabbath. His ministry ended on a preparation day, Passover. He completed the cycle. Major things happened to Christ on the Sabbath, for instance, He was resurrected on a Sabbath. Major things occurred in the history of Israel on the Sabbath as well. All those events draw attention to one supreme purpose for the Sabbath.
Jesus quotes Isaiah 61:1-2 and Isaiah 58:7. "The acceptable year" is not a time when God is acceptable to us, but when God, in His sovereign mercy, moves to make men acceptable to Him. In other words, it is an appointed extension of His grace, of His calling of men, to make them acceptable to Him. It is a time when He moves to deliver people.
More specifically, "an acceptable year" refers to two Old Testament institutions, which these people in Nazareth would have undoubtedly recognized: either 1) to the seventh year land Sabbath or 2) to the Jubilee year. If it was the sabbatical year, think about its purpose: It was given to give the land rest, to relieve it of the responsibility of growing food. The land was to lie fallow and to produce food voluntarily for the poor, for the dispossessed, and for animals. Also in the seventh year, slaves were freed and debts were remitted.
These things, plus an additional one, occurred in the Jubilee year: seized property was restored to its original owners. They may have lost it many years before, but in the Jubilee year they were relieved of the burden of their indebtedness. They were restored the ability and power, therefore, to earn money once again, since all wealth ultimately comes out of the land. This freed them of the burden that they very likely put upon themselves.
In what is Christ's inaugural address, we see that He is stating His mission, and in each point, it involves setting at liberty.