Since apparently no one under diamond knows how to play with a split pushing champion on their team, this is how it works.
What is a split pusher?
A split pusher is a champion who excels at dueling as well as dealing damage to structures. The ability to clear a wave quickly is also preferable. They will generally be a top laner although some AD Mid laners are good split pushers as well.
What does a split pusher do?
Split pushers will use their dueling advantage to bully their opposing laner off a tower or, preferably, kill them. They will then attempt to destroy the tower and continue.
But can't the other team just try to kill the split pusher?
Yes, they can, but attempts at ganking a split pusher are sometimes difficult. If the split pusher is ahead, they can sometimes win fights 1v2, so sometimes 3 or more may be sent to deal with the split pusher. If this happens, the pusher's team can take advantage elsewhere on the map.
Well, why doesn't every team split push then?
This is up to the pusher's teammates. Sometimes teams prefer to fight all the time. Sometimes teams don't know how to capitalize when the enemy team sends 3-5 people top lane.
When should someone split push?
In general, split pushing should work all the time. There are times when you really want a split pusher, but the tactic works in every game. When the other team has a very strong teamfight, sometimes it is the right decision to win by playing around them. The more objectives you can get without being in an actual teamfight, the better off you'll be.
When should someone NOT split push?
The short answer is "Whenever you don't have wards to spot enemy rotations." It is incredibly important to know when and from where the enemies are coming as a split pusher. If you see the mid laner and junger start to move through their jungle while you are at the inner top tower, you can assess how long you have and whether it is worth it to take the tower and likely die or make your escape.
What should the pusher's team do?
They should keep pressure in another lane. The moment the other team commits to one side, the other side pushes hard. The most important thing for the main team to remember is that, regardless of how good their catch is, they will likely be fighting a 4v5. Do not engage and do not get engaged on, but stay far enough up that you will get a tower if they move to the pusher.
Yes this is meant to sound condescending. It's still how you should deal with a split pusher on your team. Please learn people.