Why do books titled "Abstract Algebra" mostly deal with groups/rings/fields?

There is not that much substantial to say about general semigroups at an introductory level, in comparison to groups, say, and what there is to say at this level (e.g. the theory of Green's relations) tends to rely on a prior understanding of groups in any case. (One thing I have in mind here --- but I am speaking as a non-expert --- is that semi-groups are much closer to general universal algebra than groups, and universal algebra is not, I would say, appropriate as a topic for a first course in abstract algebra.)

The theory of groups lends itself well to a first course, because the axioms are fairly simple but lead fairly quickly to non-trivial theorems, such as the Sylow theorems. Groups (especially group actions) are also ubiquitous, and the kind of counting arguments which one can make are very useful to learn. (It is the inability to make these sorts of counting arguments that causes so many problems when you investigate semi-groups, in comparison to the case of groups.)

The theory of fields is particularly useful in number theory, and Galois theory also provides a nice tie-in with the theory of groups, which in fact served as Galois's motivation for introducing groups in the first place. The theory of fields (especially the part to do with finding roots of polynomials) is also the part of abstract algebra that is closest to high school algebra, so it is not surprising that there is a substantial focus on it.

The theory of rings normally appears both because it is a precursor to field theory (fields are particular kinds of rings, and the polynomial rings $F[x]$ also play an important role in the study of fields), and because it includes many basic examples from mathematics, such as matrix rings, the integers, quaternions, and so on. Rings also play an important role in the study of group representations (via the appearance of group rings), which, even if they don't appear in a first course, are just over the horizon.

Summary: So overall, I think the answer is that groups, rings, and fields are the parts of algebra that are most closely connected to the basic core topics of mathematics, and are also closely integrated with one another. (Groups not immediately obviously so, but because of Galois theory and group rings, for example.) The theory of semigroups, by contrast, doesn't play much role in the rest of mathematics, and the theory that does exist is more complicated than the theory of groups (despite the axioms being simpler).

Historically, the first "modern algebra" textbook was van der Waerden's in 1930, which followed the groups/rings/fields model (in that order).

As far as I know, the first paper with nontrivial results on semigroups was published in 1928, and the first textbook on semigroups would have to wait until the 1960s.

There is also a slight problem with the notion of "simpler". It is true that semigroups have fewer axioms than groups, and as such should be more "ubiquitous". However, the theory of semigroups is also in some sense "more complex" than the theory of groups, just as the theory of noncommutative rings is harder than that of commutative rings (even though commutative rings are "more complex" than rings because they have an extra axiom) and the structure theory of fields is simpler than that of rings (fewer ideals, for one thing). Groups have the advantage of being a good balance point between simplicity of structure and yet the ability to obtain a lot of nontrivial and powerful results with relatively little prerequisites: most 1-semester courses, even at the undergraduate level, will likely reach the Sylow theorems, a cornerstone of finite group theory. Semigroups require a lot more machinery even to state the isomorphism theorems (you need to notion of congruences).

Historical inertia. A relatively small number of people were responsible for more or less deciding the modern abstract algebra curriculum around the beginning of the 20th century, and their ideas were so influential that their choice of topics is rarely questioned, for better or for worse. See, for example, Section 9.7 of Reid's Undergraduate commutative algebra.