Level shifting digital signals with a comparator
Could someone explain to me what exactly is happening with the
  voltages at the positive comparator input when the a logic high and
  logic low are transmitted, respectively?


When the input is 1.5 volts, clearly and without any math, the voltage at the non-inverting input = 1.5 volts (this causes the comparator to produce a logical 1 output).

When the input is -2.0 volts use what you know about potential dividers to calculate the voltage. Basically you'll have 3.5 volts across 35k. This causes a current of 0.1mA and that current produces a volt drop of 1.5 volts across the 15k. Because one end is tied to +1.5 volts, the non-inverting input has to be at 1.5 volts minus 1.5 volts i.e. 0V. This cause a logical zero output on the comparator.

So, basically I am disagreeing with @sabirmoglad's calculation because he's saying it's the other way round but he's wrong!