Why is this clock signal connected to a capacitor to gnd?
C7 and R58 form a high pass filter, also known as a differentiator.

The purpose of using a differentiator in this spot is to cause a short pulse on the rising and falling edge of the clock signal. 

This diagram shows the effect of a differentiator on a square wave (which your clock will be.)



As you can see, it makes short pulses on the edges of the square wave.



I'm not sure why that circuit needs the short pulses instead of the square wave, though.  Too many ICs I'd have to look up to figure out what is going on.