Atmospheric pressure is an expression for the force exerted on a surface (in this case the water surface) per unit area caused by air particle collisions on the surface. In a cup of water, there is a uniform constant pressure acting on the water surface. However, upon inserting a straw and breathing in, the number of air particles within the straw decreases, in turn decreasing the pressure acting on the surface of the liquid within the straw. There is thus a difference in pressure acting on the water inside and outside of the straw. This results in there being a net upwards force acting on the liquid inside the straw. Thus liquid is forced up the straw. When we 'suck' on the straw harder, we further decrease the pressure within the straw and this thus causes there to be a larger net force acting on the liquid, thus it gets drawn up the straw much faster since larger forces result in larger acceleration of the liquid from Newton's 2nd Law: F = m*a.