To go to the mat for someone means to give them full support, which is what the speaker is saying they will do for Bugsy.
Consider the statement "Bugsy, if you tell me you didn't do it, i'll stand up for you and fight for you, no matter what elton says.". Is the statement "Bugsy, if you tell me you didn't do it, i'll go to the mat for you, no matter what elton says." false? Yes/no/unknown