Imagine you are playing a game of tag with your friends. The rules of the game say that before you can start playing, you have to give the person who is "it" a high five to show that you are ready. This is like proof-of-payment.
In real life, when you want to use a bus or a train, you have to pay for a ticket that shows you are allowed to ride. Once you have paid for your ticket, you have to show it to the person in charge (like a bus driver or a train conductor) before you can get on. This is called proof-of-payment.
If you don't have your ticket with you, or you haven't paid for one, you won't be allowed to get on the bus or the train. Just like in the game of tag, you have to follow the rules and show that you have done what you are supposed to do before you can join in the fun.