Allah has given us power over animals and allowed us to eat meat, but only if we pronounce His name at the solemn act of taking life. Muslims slaughter animals in the same way throughout the year. By saying the name of Allah at the time of slaughter, we are reminded that life is sacred.
The meat from the sacrifice of Eid al-Adha is mostly given away to others. One-third is eaten by immediate family and relatives, one-third is given away to friends, and one-third is donated to the poor. The act symbolizes our willingness to give up things that are of benefit to us or close to our hearts, in order to follow Allah's commands. It also symbolizes our willingness to give up some of our own bounties, in order to strengthen ties of friendship and help those who are in need. We recognize that all blessings come from Allah, and we should open our hearts and share with others.
It is very important to understand that the sacrifice itself, as practiced by Muslims, has nothing to do with atoning for our sins or using the blood to wash ourselves from sin. This is a misunderstanding by those of previous generations: "It is not their meat or their blood that reaches Allah; it is your piety that reaches Him." (Quran 22:37)
The symbolism is in the attitude - a willingness to make sacrifices in our lives in order to stay on the Straight Path. Each of us makes small sacrifices, giving up things that are fun or important to us. A true Muslim, one who submits his or herself completely to the Lord, is willing to follow Allah's commands completely and obediently. It is this strength of heart, purity in faith, and willing obedience that our Lord desires from us.