Fitts didn't want his secret out, and since he kissed Lester, Lester now knew that Fitts was gay. Fitts killed him so that his homosexuality would stay a secret.

Why is Lester killed in American Beauty?

As for why he was killed— Frank Fitts was in self denial all along. He was homosexual and homophobic at the same time. The only way to resolve the internal conflict was to deny his sexuality to himself and to the society . Lester knew his secret which Frank could not handle.

What is wrong with Barbara Fitts?

Barbara Fitts, Ricky's mom, is also implied to be suffering from some unspecified mental illness, echoing Ricky's own unspecified diagnosis, and she is shown to have very little agency over her own life. She seems to drift aimlessly through each day, occasionally nudged into action by Ricky or Frank.