The atmosphere is thicker (extends to a higher altitude) because a warm air mass is less dense than a cold air mass. A warm air mass will expand when heat is added. Since the air at the equator is usually hotter than its surroundings, the air at the equator expends vertically. This extra vertical expansion is why the atmosphere extends to a higher altitude at the equator.

In addition to a higher topping atmosphere, when the air at the equator is heated (from the hot surface), the air rises causing uplift. This rising air causes an area of low pressure to form at the surface as the air rises. This uplift causes air to be sucked in at the surface towards the equator. Air from the north and air from the south "converge" on the equator where this area of low pressure forms. This area of convergence near the equator is called the "InterTropical Convergence Zone" (ITCZ).

The Earth is, in fact, larger at the equator than it is at the poles due to its rotation. Although assumed to be a perfect sphere, the Earth is actual an oblate spheroid. While this is true, this is not the reason why the atmosphere extends to a higher altitude at the equator.