Why don't Catholics eat meat on Ash Wednesday and on Fridays during Lent?
Generally this is as a mark of respect and remembrance of the sacrifice made by Jesus on Good Friday. Originally only a Lenten penance for the sins of mankind this was extended to include Wednesdays and Fridays on certain rogation days with special prayers during the Penitential Seasons of Advent (prior to Christmas) and Lent (prior to Easter) which is why since the liturgical reforms of Vatican II it is now done around the time when Jesus' death is remembered.
In the Catholic religion, these days during Lent symbolize repentance days so people can contemplate their sins and ask for forgiveness. Prior to the Vatican Council changes, every Friday was mandated as a day of fasting for all Catholics unless a special dispensation was given by a Bishop or parish priest. The original practice had lost much of it's significance for many Catholics, since so many dispensations for reasons of health or travel or old age hardship, caused confusion or religious scruples among many and there was always the possibility of causing unnecessary scandal to those who knew you were Catholic but didn't know you had sought a dispensation for medical or other reasons. There were geographic areas that were blessed with an abundance of seafood that made the fast seem less than salutary for many and other areas where poverty caused an unequal hardship. The updated changes made the traditional Friday fast a more positive practice when it was made voluntary instead of mandatory except on Ash Wednesday and the Fridays during Lent.
This is a little difficult to explain in a short answer. All Fridays of the year are days of abstinence from meat in commemoration of Christ's passion and death. This has been in effect since the very early Church (we're talking the first century or the second century). Pope Paul VI revised the entire scheme of fasting and abstinence (the later now refers to abstaining from meat) in his Apostolic Constitution Paenitemini on Fast and Abstinence; but it very plainly included the abstinence from meat on every Friday of the year that was not a Feast day. He did allow individual Bishops' conferences to stipulate that, outside of Lent, the Bishops could stipulate some other penitential practice instead of abstaining from meat, but it would remain meat on Ash Wednesday, and the Fridays of Lent. Although the American Bishops have done this, the British Bishops just rescinded this as no one was following the practice of giving up SOMETHING on Fridays, so they have re-instituted the practice of abstaining from meat on all Fridays of the year (that are not feast days). Short answer? to commemorate Christ's passion and death.