What does "carry the twosie" mean in here?

"Twosie" is a diminutive of "two"; compare onesie-twosie, an idiom meaning in ones and twos, or "twosie" a type of sleepwear in two pieces. This is by analogy with "onesie" which is a better-known word for a single-piece garment originally for small children (hence the diminutive) and now used as sleep/leisure wear by adults.

"Carry the two" is an example of a procedure in arithmetic where you do multiplication (or addition) digit by digit, carrying the excess over to the next column: see this explanation on WikiHow.

So "carry the twosie" is a childish way of doing multiplication, presumably intended for humorous effect.