Consider some scenarios:
Jim is not a nice man and is very demanding and abusive and goes to vote. He wants his wife to vote for X, but he suspects that she will vote for Y. Today, she can vote for X or Y and tell her husband that she voted for Y. He has no way to prove her wrong. With receipts, he can tell her, on no uncertain terms, that she is to show him her receipt afterwards. If she doesn't produce a receipt showing she votes for X, he'll beat her. With receipts she does not have a free vote, without receipts she has a free vote.
I want to buy your vote, and will pay you €10 to vote for X. Today, you can vote for who ever you want, and tell me you voted for X and demand payment. I have no way to prove you wrong. Hence vote buying is hard to do, because you can't know if you're actually buying votes. With receipts, I can ask to see your receipt, and only pay out if you vote for X. With receipts, vote buying now becomes an actual thing that I can do.