What is the purpose of these capacitors in this State Variable Filter schematic?
They are DC blocking capacitors, to keep the whole circuit AC coupled.  In particular, they keep any DC current from flowing in the 'frequency' pots, which could cause audible scratching when the knob is adjusted.  If you look at the circuit, the only ground references are at R11 and R14.

The reason it simulates OK is probably because your simulator has all matched op-amps (you could try leaving them out and then doing a Monte Carlo simulation on the op-amp offsets), and because the sound of a DC-biased pot being adjusted isn't something you can simulate.