This relatively old expression is seldom heard nowadays. When you use this idiom to describe the apartment that your friend is staying in, what you are suggesting is that the place is very small. It is very cramped. When you say there was hardly any room in the stadium to swing a cat, what you mean is that the place was packed. It was very crowded.

You’re supposed to be the Vice President of the company. Look at your office. There’s not enough room to swing a cat.

There are several theories about the origin of this expression. Many scholars claim that the ‘cat’ in the expression refers to a whip and not the animal. The whip was called “cat o’ nine tails” which in everyday conversation was reduced to ‘cat’. It was mainly used to flog people. If you wish to hurt someone with a whip, you need to take a big swing with it; you need to take the whip back. This is not possible if the place you are in is small or cramped — if it is, it doesn’t allow you to swing the ‘cat’. Another theory put forward is that the cat in the idiom is indeed an animal. In the past, a live cat was swung by its tail, and archers were asked to shoot at it. The poor animal was used for target practice.