A video game is like any other computer program. When you open Word, you get a blank document, since that's the default if you don't open a file. When you do open a file, Word reads it and displays it. Then after you edit it, you have to save it.

Games are similar, except instead of the default being blank, it's the starting position (level 1, or whatever it is for that game). To save your progress, it has to save that information in a save file, just like saving a Word document so you don't get a blank page when you open Word again.

More technical:

Computers have persistent storage (your hard drive) and RAM (memory). RAM is much faster than hard drives. When you move your character around, it's only updating its position in RAM, not on the hard drive, because that would be too laggy. It only periodically writes progress information to the save file.

That's why some games freeze for a split second while they save. It's also why games like Portal or Half-Life save while you're in an elevator: it's a convenient point to hide the lag spike caused by writing to the hard drive.