If someone is well known or famous like a celebrity, a claim to fame is the reason behind their success. For example, Neil Armstrong's claim to fame was being the first person to walk on the moon. (I wanted to use that example because I just read he died.) The Beatles claim to fame could arguably be their famous Ed Sullivan Show performance on February 9th, 1964. The band became an instant hit after this. As they say, the rest is history.

It means like : my greatest success was...