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.
The circuit overall is a 16-byte memory bank for a homebrew computer of some sort, with manual programming capability via the switches and lights. Probably the most complicated 16 bytes of memory you'll ever see!
The R-C combination being asked about is used only when the CPU is running — i.e., when PROG is not asserted. It is probably being used to shorten the high time of the write-enable pulses going to the memory chips in order to meet hold-time requirements when the CPU is writing to memory. This only works if the clock high time is significantly longer than the R-C time constant.
10 nF × 1 kΩ = 10 µs
So presumably the clock is something less than 50 kHz.
Actually, the 74189 is not a slow part — the minimum write pulse width is a few tens of ns — so the R-C time constant could be much shorter, by a couple of orders of magnitude.