It is a pun, based on two different meanings of the word 'flush'. The flush on a toilet is the handle that causes water to come into the bowl and 'flush' out the dirty water into the sewer. A person is said to look 'flush' if their face is red and sweaty. There are other meanings too

As Bramwell said, it’s a play on words - to say someone is ‘a bit flush’ is to say they have a lot of cash at that point, as in ‘let’s go out tonight, I’m feeling a bit flush’ i.e. ‘I’ve got some spare cash to spend’. It’s a common expression in the UK, I don’t know about anywhere else. You ‘flush’ the toilet, which describes the sudden flow of water, and also if someone is red in the face and sweaty you can say that they look ‘flushed’, as in their skin is flushed with blood.