How does this timed fan controller work?
I take it you meant to write 563 rather than 653. 

Well, it would make more sense if SW1 went to live rather than neutral. 

R3 is to limit the current (if the power is suddenly applied at the AC line peak) to less than 1A, thus protecting D3 and C1 from overcurrent. 

C4 may be to kill glitches on the counter output or for some other reason to do with the triac.