Paying with Apple Pay is easy because you can simply touch your iPhone to a pay terminal and your iPhone wakes up, presenting your default card. If you wanted to open another Passbook item on iOS 8 or prior such as a boarding pass, Starbucks card, or LevelUp card, you would have to locate your Passbook app to launch it and then find your card.
But beginning with iOS 9 you can simply double-click the home button while your screen is sleeping or locked and instead of bringing up your recent app screen you’ll get your Wallet (no longer called Passbook). If you accidentally unlock your phone with TouchID you may need to double-click the home button with your fingernail.
If it doesn’t work check to make sure “Double-Click Home Button” wasn’t disabled in Settings > Wallet & Apple Pay.