Catholics abstain from eating meat on Ash Wednesday and on Fridays during Lent not because of some rule in the Bible, but because of a tradition begun in the early church. Various books of the Bible encourage the spiritual practices of prayer, fasting, and almsgiving.
It wasn’t long until Church leaders began encouraging abstinence from meat on certain days as well. (The required days for fasting (eating less) and abstinence from meat have varied over the centuries.) These practices of self-denial are considered forms of penance. They are meant to be calls of conversion, to get us back on track if we’ve wandered off from our spiritual journeys. Some people find that fasting and abstaining from meat help them to be thankful that they normally have access to meat and plentiful food. Denying themselves these things serves as a reminder to pray and work for justice in a world where so many still go hungry.