Basic operation of low pass filters?
I am having tremendous difficulty understanding how the capacitor
  steals all the current that would not otherwise leak into the output
  wire....


When ever you design a passive filter (RC, RL etc.) you have to take into account the impedance of the loading circuit and the impedance of the feeding circuit. If you don't do this you get the wrong filter performance.


  .... reproducing the frequency that entered in, only with a different
  amplitude from the resistor.


Yes, that is what happens if the loading impedance is significant compared to the filter component impedances. This is why many filters are designed to feed a high load impedance such as an op-amp. On the other hand, a passive cross-over filter inside a speaker has to deal with very low impedances (such as 8 ohm) and, as such, has to use different (and usually more complex) design equations.

Regarding the capacitor concept. If you just think of it as an impedance that lowers linearly with applied frequency, it should all make sense.