I'll be frank. Harris is not a particularly gifted author--if enhancements to writing ability emerge, he would be wise to use them. His sentences can be long and digressive, and his organization of arguments is a bit jumbled. He also has an annoying tendency to rely on citations of his past publications in lieu of developing more elaborate arguments.
Despite these shortcomings, there is much of value in the book. He does a nice job of showing that the arguments against enhancement are largely based on prejudices and intuitions that result from fear and limited imagination. His treatment of Leon Kass--if I had to choose an archenemy, Kass would certainly be the odds-on favorite, so great is my hatred for him--is pointed and downright hilarious.
His main argument--although not explicitly suggested as such--is perhaps decisive on the issue. It's essentially the same argument for a woman's right to choose abortion, or an individual's right to take psychotropic drugs in the privacy of his own home. It begins with what Harris calls "the democratic presumption":
One of the presumptions of liberal democracies is that the freedom of citizens should not be interfered with unless good and sufficient justification can be produced for so doing. The presumption is that citizens should be free to make their own choices in the light of their own values, whether or not these choices and values are acceptable to the majority. Only serious real and present danger either to other citizens or to society is sufficient to rebut this presumption. If anything less than this high standard is accepted, liberty is dead. 
As Harris notes, this is essentially the principle that Mill defends in On Liberty. It puts the burden on those who would restrict research into or employment of enhancement technologies. The issue is not, are enhancements morally right, but rather, is there a good enough reason to forbid those who judge them to be desirable from developing or using them?
This emphasis on individual choice (and parental choice in the case of germline genetic engineering and other reproductive technologies) is what distinguishes contemporary proponents of genetic enhancement from the much maligned eugenicists of the previous century. Equating the two is nothing but guilt by association.
What was problematic in eugenics is not the motive to better humanity--indeed, any society which enforces compulsory education has such a motive--but the totalitarian imposition on individuals that would compel them to undergo selective breeding or participation in dangerous experiments. Indeed, if anyone is close to the totalitarian spirit here, it is those who would restrict me from using the enhancements I desire to use on myself or my offspring.
I love this argument often employed by HE proponents (especially its nifty reversal of totalitarian roles--who's the Nazi now, Luddite?)! With the exception of certain conservatives, virtually everyone subscribes to this democratic presumption. The problem is that they are inconsistent in its application, especially when it comes to technology and HE.
On top of this central argument, Harris does an admirable job of showing how a right to individual self-determination as suggested here can be extended to include choices over offspring. The primary reasons given for denying such reproductive interventions, when teased out, amount to mere fears and prejudices.
For instance, the harm that might be caused to a child who grew up knowing they were designed is seen as compelling enough to restrict liberty. Potential psychological harm (of which there is no evidence; children of in vitro fertilization certainly seem to adept well enough despite the "artificial" process which generated them), especially when we're talking about children who are hopefully designed to be more resilient to such things, is by no means decisive.
Moreover, his consideration of the moral status of embryos is perhaps the best discussion I've seen of the subject (although admittedly I am not well versed on the extensive literature here). The most ingenious point is his discussion of embryonic splitting and recombination. An embryo of four cells can be split into four separate single cells, each of which can develop into a fetus (indeed, identical twins are the result of this kind of process). However, they can also be brought back together into a single embryo that will develop as normal.
So what has happened if we split an embryo into four and then recombine them (a process in which no matter is created or destroyed, but only rearranged in space)? Have we spawned four lives from one, and then killed off three of them (in which case we might also ask, which of the four would survive)? Applying the doctrine of ensoulment at fertilization only makes things more absurd; can souls split and recombine too? (Incidentally, even if it is the most popular view, ensoulment is a vulgar prejudice that I don't take too seriously; the kind of people who advocate it are not likely to be the kind that respond to reason.)
This example is also useful in addressing those who try to argue for value in potentiality. If each embryo can actually develop into many different individuals, do we have an obligation to split every embryo as much as possible in order to realize their full potential? Similarly, with the right technology, virtually any cell in the body could become a distinct individual (all that is necessary for this is the genome that every cell contains).
Embryos are just (collections of) cells. They have no intrinsic worth or value. Deal with it.
Lastly, I like his treatment of the issue of disability in relation to enhancement. He uses a simple distinction to show how the rights of the disabled have no moral force for preventing HE. Just because we value a certain trait in ourselves or our offspring by no means entails that we disvalue individuals who lack this trait.
For instance, if I were to lose my hearing, I would strongly desire to restore it through whatever technological means necessary. This reasonable desire is totally consistent with regarding the deaf as individuals who deserve the same legal and ethical status as anyone else. The same applies to distinctions between the enhanced and non-enhanced; a kind of egalitarianism can be maintained here, should one desire to do so.
In the end, I give Enhancing Evolution a half-hearted recommendation. Some of its arguments and discussions are excellent, but others are the usual fare of HE proponents, and all the while one has to deal with Harris's sometimes garbled prose. However, I would strongly recommend reading chapter 8 for his vicious and entertaining take-down of Kass (and Habermas, to boot!).