Why don't the antigens of blood type AB react to the A or B antibodies received in a transfusion?

A few reasons:

First, the transfused blood volume will generally be small, which means that the number of antibodies will also be small.  It's generally not enough to start a reaction.

Secondly, antibodies should really be thought of as the trigger rather than the bullet.  To get a strong immune response, the antibody-antigen complex needs to activate the B cells that are making that particular antibody.  In short, there are a whole bunch of other parts to the system that don't get transfused.

Thirdly, the transfused antibodies themselves may be recognized as foreign by the recipient immune system and cleared out.

The reason is because transfusion is done with component therapy, meaning whole blood is not transfused, just the individual components (red blood cells, plasma, or platelets). If whole blood is collected, it is then centrifuged to separate the components, and stored that way.

The antibodies are found in the plasma portion of blood, so when type A red cells are transfused, there is hardly any plasma present because it has been removed, so there is not a reaction. If you were to transfuse type A plasma to an AB individual, it would cause a reaction, because the anti-B would attach to the red cells and lyse them (break them apart).