Whether virtual particles are real or not is a moot question.
Here's the idea. In quantum mechanics nothing is really real unless you can observe it or measure it. In order to be observable, a particle has to have some minimum amount of energy for some minimum amount of time; this comes out of the uncertainty principle that says the product of those two things has to be bigger than a certain number.
So it's possible to conceive of a particle whose energy is not big enough or whose lifetime is not long enough to permit a true quantum measurement, but still both of them could be greater than zero. The world could be full of such particles, and the measurements would never show it.
Well, quantum field theory takes those particles seriously. It says they interact with observable particles, for example they make the electron which emits and absorbs them a bit heavier, and a bit more sluggish in motion, than it would be if they didn't exist.
Furthermore, QFT says that the virtual particles are the ones that carry the forces. For example with photons, the "real" photons make light, and other forms of electromagnetic radiation, but the virtual photons carry the electric force; a charged particle is charged because it emits virtual photons. And the other bosons, that carry the weak and strong forces, behave the same way. Real particles interact with each other by exchanging virtual bosons.
This is the story quantum field theory tells, and the justification, the reason you should at least consider beliving in it, is that it makes fantiastically correct predictions. That bit above where I said that interacting with virtual particles made the electron sluggish? It's called the anomalous moment of the electron, and the prediction, based on virtual particles, matches experiment to six decimal places.
For several decades the theorists and experimenters were in a kind of race. The experimenters would rack their brains and come up with a new way to measure down to another decimal place. And the theorists would grind through the next level of calculations (in the same theory mind you, this is not about epicycles) and come up with another decimal place of their own. And the two would match!
