Does anyone understand this comment about the continuum hypothesis?

You might read "if we do not allow generic functions to exist then the continuum hypothesis is obviously true" as a reference to the fact that Borel sets satisfy CH, or possibly to the fact that CH holds in Godel's constructible universe. For the "obviously false" part, I suspect a nod to Paul Cohen's comment:

A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now $\aleph_1$ is the set of all countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set $c$ is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach $c$. Thus $c$ is greater than $\aleph_1$, $\aleph_\omega$, $\aleph_\alpha$ where $\alpha = \aleph_\omega$, etc. This point of view regards $c$ as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction.

I feel disingenuous not adding that my personal view is that $\aleph_1$ and the real line are both proper classes, so that CH is not a meaningful question.

See this answer by Hamkins on multiverses and switches. What Gromov probably means is that if we allow the full power of set-theoretic methods like forcing then there is no reason whatsoever to assume that the set-theoretic universe we are working in should necessarily satisfy CH (an example of a switch in the sense of Hamkins).