It's typical American English — we generally don't use the word "till," and we often say "register," since "cash register" is too long. It would be more accurate to say "paying for the clothes at the register," although it would be possible to say "buying" when speaking off the cuff.

I agree with Gary that probably means that the person is paying for their clothes at the checkout. 'Cash register' is an old-fashioned term for what can also be called a 'till'. This is the machine for registering purchases and taking payment in a retail outlet. 'Cash register' is very rarely used these days, and - as far as I know - it is never abbreviated to 'register'. 'Register' on its own does not suggest a place to pay. I suspect that this was written by a non-native.

It is probably short for 'cash register', which is also known as the till. That is the machine that the assistant puts the money in when you buy something.

Just to reinforce: Phil is absolutely correct. Register is still very common US usage.