Why does air subside in the Eye - the lowest pressure area of a tropical cyclone?
I will just talk about an already formed tropical cyclone. Converging winds spiral in (counterclockwise in northern hemisphere) over the warm ocean waters towards the central low pressure area of the eye. 

At the eye they spiral upwards, taking the warm, moist ocean air high into the atmosphere. As it reaches cooler elevation, the air releases its latent heat, adding more energy to the storm. 

So you have a tremendous volume of air converging from 360 degrees around the hundreds of miles of the storm, all converging on the eye and spiraling up with an intense updraft, creating a very low pressure near the surface in the eye/core. 

High above the eye wall this upflow begins to spread out. This permits cool dry air above the eye to sink down into the central core of the eye (that is why the eye is often clear and cloud free). 

So why is the hurricane core/eye low pressure when there is cool dry air subsiding down into the core/eye? 

Soundings taken within the eye show a low-level layer (at surface) that is relatively moist with an inversion above, i.e., it appears that the sinking cool dry air (which warms somewhat as it is compressed in descent) never reaches the ocean surface, but only reaches within 1 - 3 km (1 - 2 mi) of the surface. See, among other sources, http://www.aoml.noaa.gov/hrd/tcfaq/A11.html.