Algebra and programming are very, very similar. Algebra [1] deals with formal languages and their rewrite rules in general. Programming is dealing with a certain (but yet very wide) class of formal languages and an extremely wide variety of rewrite rules (i.e., algebras).

It does not matter that nobody is using, say, a term "refactoring algebra", but it is still, essentially, an algebra.

[1] Here I'm referring to a generic meaning of "algebra", as in https://en.wikipedia.org/wiki/Universal_algebra

Even if it was just that, it would probably work as a predictor of skill. After all, people being good at one system of formalized abstract thought must be a reasonable predictor that they will be likely to master another one.

But it's more than that. Algebra and "algebraic reasoning" (to use a non-technical term I just made up) are the underpinnings of Computer Science, which is why Algebra is featured prominently in the CS syllabus. Of course this is obvious in CS theory (where you'll play with numbers as if they were programs), but it's also useful in order to write actual programs.

I'm not saying you must understand Algebra in order to write a CRUD in your typical web application. Likewise, you probably don't need to understand Relational Algebra to write SQL for the aforementioned web application. You can understand surprisingly little in order to "muddle through". And intelligent self-made programmers will actually be able to write complex software without a solid theoretical foundation to back them up (maybe rediscovering the wheel along the way). It's definitely possible to write software without knowing the theory.

However, I think you'll be a better programmer if you do. (My highly subjective opinion: it's definitely more fun if you also understand at least some of the theory!)