Faddeev–Popov ghosts in particular are artifacts of covariant quantization in the path integral formalism. One way to think of them is as a kind of Jacobian determinant of the transformation to physical degrees of freedom.

The two types of "ghosts" here are very different. In both cases, though, they're mathematical artefacts rather than anything "physical", but I'll try to explain them as well as I know. Disclaimer: The most advanced physics I've done was a first course in this stuff, so I might be wrong about some things.

For what they're calling "Higgs ghosts": There are two different ways of describing the electromagnetic and weak forces, and which one is best depends on how much energy the particles you're dealing with have. At really high energies, it makes the most sense to talk about a combined "electroweak force", described in terms of four fields with 2 components each (often called W1, W2, W3 and B), and one 4-component field (the Higgs field).

In contrast, at low energies, it makes more sense to talk about the electromagnetic force, with one 2-component field (y), and the weak force, with three 3-component fields (W+, W-, Z0) and a 1-component field (the Higgs field, again). So, where did the other three components of the Higgs field go? Well, we just rearranged things - if you check, the total number of components stayed the same. There are various names for this rearrangement, and I haven't seen this one before, but I guess they're calling these "missing" components "ghosts".

As for the other type, the Faddeev-Popov ghosts, those are more obviously mathematical artefacts. Normally, you'd start by writing down a "physical" Lagrangian ("physical" here meaning something like "written in terms of actual physical fields").

But it turns out that you can't actually calculate with the physical Lagrangian. So you have to rewrite it in a (mostly) mathematically-equivalent way, which involves extra fields. These fields come along with extra rules which basically say "no state you can actually measure involves the ghost fields in any way". Really, they're just there as a calculational aid and aren't physically "real", and they're called "ghosts" to reflect that.

Hope that's at least vaguely comprehensible, it's difficult to explain this stuff without assuming a lot of background knowledge.

Mostly right, but one correction:

You actually _can_ calculate with the physical Lagrangian. This is what lattice gauge theory simulations do. But it's inconvenient and difficult in perturbation theory, so physicists use the Fadeev-Popov ghost trick instead. The resulting computations are _entirely_ mathematically equivalent, not "(mostly)".

It's the same as noting you can more quickly calculate 19x21 by noting it's equivalent to 20x20 - 1. This ghostly term is the -1

It doesn't really make sense to ask if the ghosts were predicted. They're mathematical artifacts of a particular computational formalism. There are other computational formalisms for QFT which don't feature ghosts, and which give the same answers for any observable quantities.