The point is that reaction in question is thermodynamically favored to happen in a certain direction, whether or not the enzyme may be present, and that fact, rather than the presence of the enzyme, is the explanation why it happens in that direction. An enzyme can lower a temporary hindering or block to the reaction, for example, by lowering a potential barrier. So the enzyme is like something that turns on a tap for water to flow out of a tank. The tap temporarily holds the water back, but the stored potential energy in the water reservoir is the reason for the water's flowing when the tap is openned.
What Feynman is referring to here, I believe, is the fact that for all fundamental building-block physical theories, such as electromagnetism, quantum mechanics and Newtonian mechanics, any solution to the relevant equations can be reversed in time to find another solution. That is, fundamental physical laws don't know the difference between past to future and future to past time progression: they work in exactly the same way in both cases. The same holds for general relativity although it is less relevant there because people most often tend to think of the spacetime manifold rather than time evolution. So if the enzyme "cared" or "knew" which way the reaction were going, its behavior would tell against at least one of these fundamental theories of physics which are all reversible.
I say more about these issues in this answer here to a related Biology SE Question.