How does evaporation even possible?
I remember my days when I started confusing myself in the exact same way. The issue I believe is that you haven't been explained the details of how all this happens.
Evaporation:
All molecules in a liquid have various amount of energy. But the ones at the top, the ones that form the surface - they have the maximum energy in them because they are present there by overcoming any forces of attraction. (The other way to imagine this would be, the layer of molecules at the surface will have mostly downward forces of attraction on them compared to the distributed force of attraction from all directions on a bulk molecule.) Now these surface molecules have the tendency to escape. These molecules will individually leave the liquid by getting rid of the forces of attraction because it is easier for them compared to bulk molecules. And when individual molecules leave the liquid, it is called evaporation. Although this is a form of vapourisation, it doesn't need a constant temperature where the change of state occurs. They are just individual molecules that leave the liquid and escape and cause a vapour over the liquid.
Boiling:
But in boiling, a lot of constant heat supply, constant temperature, constant pressure. All this is required. And when this is obtained, the liquid changes its state and becomes a gas. This gas forms bubbles and escapes from the bulk of the liquid as well and not only from the surface as in case of evaporation.