Pulling a phone out and putting it up to a reader takes less time than pulling a wallet out, pulling out the right card, then swiping it. It also takes less space to store cards electronically vs. physically in a wallet, especially if you have a lot of store cards and loyalty cards.
ApplePay also has the advantage of using a one-time credit card number so it's more secure (your credit card number would have never been hacked at Target if ApplePay was supported and used), etc.
The thing that sucks about ApplePay is that about 1/2 of my credit cards (and all loyalty cards) don't support it, and there's only a few retailers where I can use it.
I love ApplePay when I can use it, though!