I guess I understand the main joke correctly — putting your right hand into the instrument’s bell is a technique used with the horn (French horn), and this guy has carried over the habit into his playing the trombone. But why is he at a doctor’s? Because it got stuck? Did he do that while playing? No, you couldn’t reach — so it was from the minute he picked it up? Weren’t there resources to go to before a doctor? The players in the low brass section of whatever ensemble he plays with?
Anyhow, let’s not let the horn theme pass without checking in with Flanders and Swann: