It's like two footballers on the same team running down the field passing the football back and forth. The two footballers represent two objects attracted by a force and the football itself represents the exchange particle. So, say we have an electron. These photons travel a certain distance after which, if they haven't reacted with anything, they cease to exist. In fact, we can say that they never existed at all - they were virtual particles.
Gravity is unusual.
Einstein revolutionised our views of gravity and his theories didn't need exchange particles.