Actually I don't think Feynman is referring to the second law. Although he is wandering some way off topic in this piece, I believe he's referring to 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.
Imagine a glass falling from a table, hitting the floor and shivvering into thousands of shards. That process conforms with all the fundamental physical theories that govern it: the mechanics of motion and the solid state physics that governs how objects stay together as a whole. Now imagine that we could freeze the time evolution, and change the motion state of every glass shard flying off the impact so that its velocity vector were pointed exactly in the opposite direction, before letting the physical process keep going. You would also have to reverse the direction of the wavefronts of all the acoustic waves travelling off into the floor and air as well as the wavefronts of the electromagnetic radiation produced by the impact. We would see all the shards coming together at the impact site, the glass reassembling itself (the edges would line up so that the chemical bonds would all reform) and jumping back onto the table unblemished! This, believe it or not, is a valid solution to all the relevant equations of physics here, just as valid as the one where the glass falls and shatters.
Feynman here is alluding to something in thermodynamics known as Loschmidt's Paradox: the problem of explaining why physical processes seem to have a preferred direction when the underlying laws are time-reversible. The answer is the second law of thermodynamics whose modern accepted explanation seems to be that the universe, somehow, began in an exquisitely low entropy state such that any "random walk" in phase space undergone by a physical system will increase the system's entropy with overwhelming probability. The same would not be true if the universe began in or near a maximum entropy state. This leaves the mystery as to why the universe began in that state; a highly readable account of these ideas is to be found in:
Roger Penrose, "The Road to Reality: A Complete Guide to the Laws of the Universe", Chapter 27 "The Big Bang and its Thermodynamic Legacy"