Why would time go slower the closer you get to a black hole?

Mathematically, it's because gravity is a distortion in four-dimensional space-time, causing the space coordinates to "slop over" onto the time coordinate.

Another way of looking at it is "gravity slows time". As you move deeper in a gravitational potential, time moves more slowly. A black hole has an enormous gravitational field.

Finally, as a photon ascends in a gravitational potential, it loses energy. This shows up as a gravitational red shift. This has the same appearance as the light source vibrating more slowly, as if time were slowing down.

One of the interesting bits of relativity is the "equivalence theorem". You could also call it, "what you see is what you get". A space craft under acceleration looks the same as a space craft sitting in a gravity field, so relativity treats them the same. Gravitationally red-shifted light looks like it's coming from a source where time is moving more slowly, so relativity treats it the same.

mass is increasing, therefore speed is decreasing.