I'm on Chapter 4. I'm continuing even though Lewis has instructed me to stop reading. (He says anyone who doesn't agree with his first argument should give it up.)
It boils down to this: because most people feel some sense of fairness and overarching morality (not only "I don't want you to do that" but "You shouldn't do that - it's wrong."), that means there must be an immaterial Lawgiver who put a sense of Moral Law in each of us. I think all of this is perfectly explicable by evolution and socialization. And supported by the indications we see that other animals have "moral codes," such as "Don't eat before the higher-ranked wolves" or "Don't have sex with anyone but the Alpha chimp, unless you can be really sneaky about it" (guilt!).
He also completely overlooks the selfish value of benefiting society. He starts well, saying, "[Human beings] see that you cannot have any real safety or happiness except in a society where every one plays fair," But three sentences later has completely forgotten this concept, and says it's silly to say it's good to benefit society, because wanting to benefit society is unselfish, so it's just begging the question. Except he started the conversation with an admission that helping society helps the individual!
He also misses the fact that human behavior all takes place in roughly the same environment, and this was probably even truer when evolutionary pressures were at their greatest, so it's not a supernatural-level surprise that we are hard-wired and socialized via long tradition to adopt similar cooperative behaviors. It becomes a (granted, complex) series of "if-then" statements: If no one in a clan can trust each other, they fail to cooperate and all die; If most people in a clan feel significant psychological pressure to be trustworthy, they can cooperate and survive.
If Lewis considered engineering, one wonders if he would find an extra-universal entity that bestows the Law of Design. "Look, all people throughout history have made boats that displace more water than that equal to their weight. Clearly this means there is a God of Boat Design!"