Why A digital filter works by taking the incoming stream of samples, delaying copies of it by one sample period at a time, multiplying each delayed copy by a coefficient, and summing the results. Each of those delay-and-multiply stages is called a tap, so the taps are literally the incremental delays the filter algorithm operates on. More taps means a sharper, more selective response, at the cost of more computation and more delay (latency) through the filter.
Watch out The debugging answer confuses a DSP tap with a hardware test point or a 'tap' on a transformer winding; in DSP the term refers to the delay stages in the filter structure, not to access points.
Tap = delay stage. More taps, steeper skirts, more latency.