All of matter is made of particles. In a liquid, the particles move are moving about randomly. Some of these particles are moving fast enough to break free of the surface. They form a vapour above the liquid.
As we heat the liquid, more particles can break free producing more and more vapour. This vapour will produce a pressure above the liquid. (In our picture, there is a lid of the container and you can think of the vapour pushing up on this lid).
Eventually, the pressure from the vapour will equal atmospheric pressure. This is the boiling point and, as we continue heating it, the liquid will turn into a gas. I.e. the liquid starts to boil when its vapour pressure equals the surrounding pressure.