Combinators are math, and a little like Lisp - building functions from primitives and operations with the ability to apply them, where even the notion of variables are functions - functions all the way down.

The y combinator is this: λf.(λx.x x)(λx.f(x x))

Lambda diagrams get you visualizations like this:

https://tromp.github.io/cl/diagrams.html

When considering logic and functions, when thinking in the space of combinators, you can ask questions like "What is Plus times Plus" and have a sensible result. https://www.youtube.com/watch?v=RcVA8Nj6HEo

Combinators are awesome.

The site linked by OP is a specific collection of combinators with bird names, riffing on the "To Mock a Mockingbird" puzzle book and subsequent meme of giving combinators bird names.

Or better yet, the y combinator is this: W S (Q (S I I))

The whole point is that we don't need no stinking variables.