Because we do not wish to harm any other living being. You could also look at it from the perspective of karma – what we do to others comes back to ourselves, either in this life or in the next.
Those who eat meat run the risk of becoming animals and be slaughtered in their next life, as a reaction to their participation in the slaughter of animals in this life, which in itself is a good reason for becoming vegetarian. You may wish to read further about the advantages of being a vegetarian here.
In reality though, we are more than vegetarian. Our principle is, that we only eat food that has been offered to Krishna first, and Krishna only likes food that is either vegetarian or comes from cows milk.
The real reason we do not consume meat, fish, or eggs, is, that they are items Krishna would not wish to receive as offerings.