Apple Pay can also be used within apps on devices that support Touch ID, which includes the iPhone 6, the iPhone 6 Plus, and Apple's upcoming iPad mini 3 and iPad Air 2. Apple's iPhone 5s does not appear to be compatible according to the Apple Pay website, even though it has Touch ID.
Wow, this particular example of Apple artificially making hardware obsolete is pretty bad. Worse than Mac Mini 2011 and MacBook Air 2011 owners not being able to use Handoff, despite having the correct and necessary Bluetooth 4.0 hardware.
The iPhone 5s doesn't have the "Secure Element" that is required to store your payment credentials:
With Apple Pay, instead of using your actual credit and debit card numbers when you add your card, a unique Device Account Number is assigned, encrypted, and securely stored in the Secure Element, a dedicated chip in iPhone, iPad, and Apple Watch.