You can only "see" things by receiving pulses of light. As an object crosses the event horizon, a distant observer sees the light from the object get infinitely redshifted. Effectively if you imagine light wave as discrete pulses, then the time between each of those pulses gets larger and larger (i.e. the frequency of the wave gets smaller, tending towards zero --> redshift). In my opinion, thinking in terms of redshifts is the cleanest way to think about all instances of time dilation, since that is what we observe with telescopes
So the distant observer takes infinite time to see the object cross the black hole's event horizon, because the light from the object that reaches the observer gets infinitely redshifted (there's a relationship between properties of spacetime and redshift that allows you to calculate this)
