Let's see how much I can extract from that description... (Okay, after having written this comment, I can say it's "not much", but at least I think I can identify which parts are more glue and reshaping than contain real content. I won't be upset if you hit that [-] to minimize this comment ;) )

> the monoid of continuous endomaps

An "endomap" is a functions `T -> T` over the same base type `T`. You might call these "unary operators".

The "continuous" adjective means that `T` is some kind of topological space -- which agrees, thank goodness, with the "topological" we see earlier in the quotation -- and that our functions preserve the extra properties possessed by these spaces. For the most part, that means that points have some sense of connectivity and continuity (though not necessarily "distance").

The "monoid" bit here doesn't actually add much; it's just highlighting some behavior that unary operators already have. Namely, you can compose two unary operators in sequence, and there's an operator that returns the output unchanged. There's some extra color around making sure the composition is continuous, but we can take the author on faith for now that things work out.

> the one-point compactification of the discrete natural numbers

This is the concrete `T` above.

The "natural numbers" are hopefully clear; we can probably assume the author is including zero, but I don't think it actually matters. I think we just need a set of countably-infinite cardinality.

Since we're talking about continuous maps, we need to consider the natural numbers as some kind of space, with some kind of concept of continuity. The "discrete" qualifier tells us that every natural number is its own island. (This is why I don't think it matters whether we have zero or not.) Any set of elements can be given the discrete topology, so it's not very interesting in its own right. Usually we're interested in what other mathematical widgets do at these boundary cases: since they're effectively contentless, we can better understand what those other widgets contribute purely on their own.

The "one-point compactification" of a topological space is a lot harder to dissect. In a very small nutshell: some subspaces of a topological space can be "compact". If the whole space itself is not compact, there's often a way to turn it into a compact space by adding a single extra point. Compact spaces have some really nice properties, so if you have a space that isn't compact, sometimes it makes sense to move into a compactification of that space instead. But I don't understand this particular construction yet, so I can't explain concretely what space we end up with.

Unfortunately, that also means I have no intuition for what maps `T -> T` we end up with on this compact space. That seems like a pretty key bit of information for the rest of the construction.

> take the monoid of ... considered as a category, then take sheaves for the canonical coverage

For the "considered as a category" part, every monoid is a category in a boring way. It's even more boring when your monoid is already made of maps `T -> T`: the one object of your category is `T`, and the arrows are the maps. This is purely a "perspective-shifting" statement, and the only reason you'd do that is to say something else that's easier to say in category theoretic terms.

As for "take sheaves for the canonical coverage", I'm not even going to try to dissect that. It looks like a coverage is one way to define a topos, and Grothendieck toposes are defined by sheaves on a site, but we're rapidly falling down a rabbit hole here. Summary: they're building a topos in an apparently standard way from the monoid of endomaps we've built up to this point.