"I have a split personality," said Tom...
"I have a split personality," said Tom, being frank.
This is a pun. The joke here is centered on the word 'Frank'. Frank can be an adjective which means "open, honest and direct", but it can also be a person's first name, and if Tom has multiple personalities, then he could be both frank (as an adjective), and Frank as a proper noun.
This joke would also work with the name Earnest, and is also the lynchpin of the play "The Importance of Being Earnest."
I suppose that it also bears mentioning that this is a joke about mental disorders.
I find this joke to be highly amusing.