He is bragging that he is "sick" with the money, meaning that he has a remarkable ability to earn it. Since he is "sick with the money," he then makes a pun about it (the "cough" and "sneeze" lines; people cough and sneeze when they are sick, of course).
It's a play on words. Sick means crazy cool or insane meaning he has a lot of money. Sick also means like you're sick, you got a cold so he's got such a bad cold that he coughs up a whole stack of money and sneezes gs (grands... $1000) so lots of money.
He has so much money that it has infiltrated his body like an infection. When he coughs, a stack of dollar bills comes out of his mouth, and when he sneezes, thousands of dollars are expelled from his nose/ mouth.