How does this Current Comparator Circuit Work?
The schottky diodes are obviously for minimizing slew delay but how exactly do they do this?


It takes time for an op-amp to swing its ouptput. This time is roughly proportional to the delta-V of the swing. So if this delta-V can be limited, the swing time is also limited. A 'standard' comparator would use a resistor in the feedback path. This requires a large output voltage to produce a large feedback current. The diodes can provide a large current with only a small (output) voltage, hence they limit the output delta-V but still produce enough feedback current.


If the non inverting input is tied to ground and both the input and reference are connected to the inverting input, how is the current threshold implemented?


The op-amp responds to the inverting input being either (slightly) below or (slightly) above the non-invering (grounded) input. If you consider the non-inverting input as grounded (basic negative-feedback opamp assumption), then the zener and its resistors provide (feed) a small current. When the input sinks (slightly) more than this current, the inverting input will be (slightlyt) below ground and the op-amp will trigger.


How is it able to measure and compare the current to a threshold?


It essentially adds the reference current (from the zener) and the input current. The circuit acts on which of the two is larger (which results in the inverting input being either (slightly) below or above ground.