As Jonathan states below, you can still perform your checks against the statements. You have the option of getting a paper receipt, but you also have a secondary backup as the wallet app will provide you with a list of your recent transactions within the app.
I use it to avoid having to hand my card to people and having someone see the number. As an employee within the credit card industry we are warned all the time of ways in which people steal credit card numbers. Just removing your card from your wallet in the presence of other humans (or even security cameras) puts you at risk of having the number stolen.
"What about RFID scanners" one might ask... Well, first off, it's NFC, not RFID, meaning the range is greatly reduced so someone would have to be standing right next to you with a scanner practically touching your phone to get the number. And lets say invisible man does so.. now what.. he's got your digital card number right? Wrong. He has a useless card number. When you register your card with Apple Pay, the app uses an algorithm that takes the card number and the device ID and creates a unique card number that will only work from that device. Add a card and look at the info within the app... There is the last four digits of your card number and the last four digits of a "Device Account Number". The Device Account number can only be used on that device, any attempt to create a fake card or add it to someone else's wallet and the transactions will fail.
In short, it's more secure. Someone stealing your card has free reign to buy whatever they want until you notice it missing and report it to your bank. Someone stealing your iPhone, will also need to steal your fingerprint to use the card on Apple Pay. And someone stealing your Device Account Number has just wasted their afternoon trying to figure out how to do it without you noticing.