It's really quite simple. It's just a matter of the fact that we can change measure on the stochastic volatility while not changing the fact that the stock is a martingale. Once we can do this, we have payoffs that have different values under different measures, so the market can't be complete.
For clarity, just consider a stock S, a money market account M and a Brownian motions B and W. Let dS = a dt + v dB, dM = r dt, and dv = b dt + w dW, where our filtration is generated by B and W, and a, r, b, and w are adapted processes. Using M as numeraire (i.e. - divide by M so as to express all prices in terms of shares of M or equivalently, assuming interest rates are zero), then under an equivalent martingale measure, assets and admissible strategies are martingales, so now S is a martingale, so dS' = v' dB', and dv' = b' dt + w' dW'.
The point is we can change measure to add a drift to W. Since dS' = v' dB', this doesn't change the fact that S' has zero drift, so S' remains a martingale. But it does change the distribution of v', and thus the distribution of S'.
If the market was complete, all contingent claims on S would be replicable. Their prices would be equal to the initial value of the replicating strategy, and hence fixed. However, since the distribution of S' changes when we change the drift of W, there are contingent claims on S which will have different values depending on the drift we add to W. Hence the market is incomplete.