Ignore the break in the middle for he said, and this is the full quote:
“I’m the only one to separate Siamese twins, the only one to take out half of a brain, although you would think, if you go to Washington, that someone had beat me to it.”
Dr. Carson launches his sentence with a description of his most famous achievement:
I'm the only one to separate Siamese twins
He then narrows this description by focusing on the most delicate and difficult piece of it:
[I'm] the only one to take out half of a brain
Note that he recasts what he actually did—separating two physiologically conjoined brains—as dividing one brain into two halves and taking one away, leaving only half a brain. This sets up his joke:
you would think, if you go to Washington, that someone had beat me to it.
Examining people in "Washington" (that is, people in the government) would lead you to think that "somebody beat me to it" (that is, somebody achieved the medical feat of removing half a brain before I did)—because people in the government obviously have only half a brain left!