How does the capacitor C1 provide filtering in this second order active low pass filter?
Well you could always go though a metric ton of algebra and get the exact equation, but I'm guessing you're looking for something a little more intuitive.

If the voltage at Vin is a sinusoid (say sin(t)), then the voltage at V+ will lag by 90 degrees (i.e. -cos(t)) because of the parallel capacitor (to ground).  Of course this produces an output signal with the -cos(t) phase.  The voltage across C2 will be the difference between these two signals, thus the amplifier injects an out-of phase signal into the input, reducing the amplitude of the resulting signal at V+.