Why does fuel vaporize in the carburetor?
Vaporization is closely related to the concept of volatility, which in turn is directly tied to the vapour pressure of a substance.

The vapour pressure of a substance is a function of temperature, and is the pressure at which a room filled with some liquid (or solid) and some gaseous parts of the substance are in exact equilibrium. Increase the pressure, and some gas will want to condense; decrease the pressure and some liquid will want to evaporate. For fuels, this is in the order of tens of kPa (e.g., the Reid vapour pressure of avgas, which is the vapour pressure at 37,8°C, is ~38kPa).

If we assume that air is an ideal gas, we can say that liquid fuel and gaseous (evaporated) fuel are in equilibrium when the partial pressure of liquid fuel equals the vapour pressure of the fuel. Since the partial pressure of the constituents of a gas are directly tied to the molar quantities of a gas, we can easily extract the fuel-to-air ratio. Given that the partial pressure of oxygen is about 21kPa (which will only decrease if you add fuel, assuming this happens with constant atmospheric pressure), we can easily see that if you leave a tank of air and fuel (with a vent to keep it at atmospheric pressure) long enough for the fuel to reach equilibrium, you have a very rich mixture indeed.

So, we limit the amount of fuel entering the carburetor to close to stoichiometric (exactly enough oxygen and air). As per above argument, all of this fuel will eventually evaporate as the vapour pressure is much higher than the eventual partial pressure reached by the gaseous fuel. The only question that remains now is how fast we can evaporate all this fuel. This only happens at the surface of the liquid fuel, so the trick here is to maximize this surface area. This is done by spraying the fuel into tiny droplets, each of which has a large surface area compared to its volume (remember, area scales with the size squared, while volume scales with size cubed - if you make the droplets smaller, the volume will decrease more rapidly than the surface area). They will thus be fully evaporated before they reach the actual cylinders. The smaller the droplets, the quicker this happens and the better mixed your fuel will be into the air.