Samsung Pay is just what it sounds like: A payment system for Samsung phones. It allows you to load up your credit/debit cards (from supported banks) and use your phone to pay. It can either use NFC terminals or 'Magnetic Secure Transmission' (MST), which is actually where Samsung Pay 'shines' since it's the only manufacturer that currently licenses the technology. MST allows the phone to mimic the magnetic strip on your cards, so it works with any regular 'swipe to pay' terminal (won't work with terminals that need the card to be inserted, like in ATMs or gas stations since the magnetic reader is too far contained for the phone's transmitter to reach). Just tap your phone and pay, just as if you had swiped your card.
It needs access to Phone because it can place service calls to banks depending on the cards you add and because it needs that for verification purposes (some banks require this as a 2-factor authorization).