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Two volatility processes yield a higher flexibility of the model. This is of greater importance if one tries to price derivatives with different maturities in one single model. A additional volatility component helps to capture the term structure of volatility, which can depend greatly on time to maturity. See for example the VIX term structure from CBOE: ...

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I think,the additional volatility factor,$v_2(t)$, provides more flexibility in modeling the volatility surface.We know $\rho$ controls the slope of the implied volatility.In the single-factor Heston model, $\rho$ is constant over maturities,In deed $$Corr[{dS}/{S\,,\,dv]}\;=\rho \,$$ which means that model has trouble providing an adequate fit to market ...

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Loosely speaking, it can be seen as inserting an additional degree of freedom in the underlying's dynamics. This can be useful from a static perspective: with an additional lever to play on, one can hope to better capture the short term implied volatility smile, which "naive" stochastic volatility models (single volatility factor, no jumps) are known to be ...

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