to understand this you kinda need a little bit of knowledge about equilibrium.
The esterification reaction is reversible. In fact, the reaction never occurs one way, it is a constant dynamic of forward reaction and backwards reaction. The point where concentration stabilizes and reaches an equilibrium is when the rate of forward and backwards reaction are the same (and this rate depends on an equilibrium constant, which is related to collision between molecules and activation energy and all that jazz).

As a dehydrating agent, water is removed from the right hand side. That is, the amount of water available for the backwards reaction is reduced, hence the number of collisions between ester and water is reduced, hence the rate of reaction is less. Therefore, there is a net increase in forward reaction rate.
