What does Mark 15:6 mean?

The "feast" encompasses the Passover, which is this day, and the week-long celebration called the Feast of Unleavened Bread, which starts in the evening (Leviticus 23:5–8). The Passover directly commemorates the tenth plague in Egypt, during which God killed the first-borns of the Egyptians and their animals while sparing the Israelites who had covered their doorposts and lintels with a lamb's blood (Exodus 12:1–32). The Feast of Unleavened Bread further celebrates God's deliverance of the Israelites from slavery in Egypt (Exodus 13:3–10). Because the two events are right after one another, they are often referred to together.

The tradition mentioned here, of releasing a prisoner at Passover, is not well documented. There is a vague reference to it in the Mishnah Pesachim, written between AD 190 and 230, which has rules about the Passover. Section 8.6 says, "[In the case of] …one who has been promised to being released from prison…we may slaughter [a Pesach sacrifice] for them [to eat from]." It goes on to say that the lamb must not be for the soon-to-be-released prisoner alone, presumably because if he isn't released the sacrifice will be disqualified. But when deciding how many lambs to prepare, the family may account for him. This suggests that prisoners were released on Passover often enough that the Jewish leadership felt they needed a standard for their sacrifice.

"Asked for" is putting it politely. Roman rulers may release a prisoner if a mob gathers and demands strongly, usually with shouting. This is, indeed, what the crowd of Jews do. Pilate is not offering this as a gesture of mercy; he's negotiating with a mob.