All the posters have made pretty good points, but I think I have the answer that you are looking for.
Vapor pressure is defined as the pressure exerted by the gas above a liquid in a closed container (where there was vacuum initially). The gas is, of course, the same substance as a liquid, and this is the only gas in this container. This means that (for sufficiently volatile liquids), some of the substance is in gas phase. This is because there is always an equilibrium between liquid and gas, written as
A(l) ->/<- A(g).
Where the position of the equilibrium lies varies with temperature, with higher temperature favoring gas. Remember that one way of measuring concentration of gas is pressure. If you think about the above as a reaction, you can use Kp = P(gas) to assign a value to the position of the equilibrium. But P(gas) is precisely the definition of vapor pressure, so vapor pressure is really an equilibrium constant for the phase change between the liquid and the gas form at a specific temperature.
If you increase temperature, the equilibrium constant, P(gas), and vapor pressure (all equivalent) increase. In a closed container, that simply means that more of the liquid will be converted to gas. In other words, a larger portion of the substance is in the gas phase as temperature is increased. At some point, as vapor pressure becomes too much, the container might break, or we might reach a supercritical fluid. Now imagine, though, that we are not talking about a closed container but an open container. All open containers are exposed to the atmospheric pressure. Atmospheric pressure is the pressure of all the gases above the liquid. The vapor pressure of the liquid (the pressure due to the gaseous form of the liquid) cannot be higher than atmospheric pressure, because now vapor is free to escape. At a temperature higher than the boiling point of a liquid, the vapor pressure is supposed to be higher than atmospheric pressure, which is not possible in an open system. Hence the liquid cannot go to a higher temperature than its boiling point, because that would imply that it is in equilibrium with its vapor, whose exerted pressure is greater than the atmospheric pressure. At this temperature, then, all of the liquid boils (bubbles form when this happens rapidly).
This also explains how evaporation can occur. If a cup of water is exposed to air, then it will eventually evaporate away, even at room temperature. The equilibrium between liquid water and water vapor favors liquid water, and the vapor pressure at room temperature is quite low. However, it's not 0, so there is some amount of vapor in equilibrium with the liquid. This vapor is free to escape, but if it does, then there is now less vapor in equilibrium with the liquid. Per Le Chatelier's principle, more liquid needs to evaporate to reach equilibrium, which results in more vapor escaping, and so on. Think of boiling and evaporation as an equilibrium reaction in which the product is continuously taken out of the system.