How does the PRESENT cipher's permutation affect all rounds when it repeats after every three rounds? I heard that the permutation in PRESENT affects all bits after only a few rounds. However, when I looked at it the bits return to their original position after only three rounds. For example, in round-1 bit b1 ---> b16, in round-2 b16 ---> b4, and in round-3 b4 ---> b1; thereafter the cycle repeats. This is the same for all bits, except b0, b21, b42 and b63 which do not move at all.

So my question is: this being the case, how does the permutation cause each bit to affect every other bit? 

I am sure I have missed the point, so a simple 'numerical' example would be much appreciated.
Yes, you might think that the bits return to their original positions.  But they don't because those aren't the bits you're looking for.  They've changed.  Specifically, you need to consider the effect of the s-boxes.  If a single bit goes into a s-box, it's very nature means that on average, two bits come out.  Similarly, if three bits go in, two bits still emerge on average.  This is the substitution component of PRESENT.  What you've focused solely on is the permutation component.  To achieve full and proper diffusion of bits throughout a substitution /permutation network (SPN), you need both components to work together.  

This repetitive substitution and permutation (what Shannon termed confusion and diffusion) creates an avalanche effect through the internal state.  If such a network happened to have the same number of s-boxes as their width (ie. 8 boxes of 8 bits), you would achieve full avalanche effect in two rounds.  If the ratio is different like PRESENT's 4/16 construction, full avalanche won't occur till more rounds happen.  It's a function of the ratio.  I don't know how many rounds PRESENT needs, but it clearly happens way before the total 31.  These networks tend to have a very generous and conservative round count to help security.

So it's the effect of the s-boxes in conjunction with the permutation that ensures all bits affect all other bits.  The diagram below is for a generic SPN but illustrates a bit's travel through it:-



One bit went in and as expected on average 8 bits come out.  It simply doesn't work if you just consider the permutations on their own.