A quick question indeed, but my answer turned out a bit wordy!

TL;DR: Yes, copy and paste the licence into a licence file—but also have somewhere in your documentation, or in the files themselves, where you say “<file> is under <licence>”.

The full version…

The organisations that publish these licences have more specific guidance, and I’m no lawyer, so I recommend you look at what they have to say. But based on experience and what I usually see, you need two things: a licence statement and the licence text.

The statement is somewhere where you tell people that this file is under that licence. For text files, like source code, you can put the statement in the file itself, usually in a comment. For non-text files (or text files that don’t support comments, or for short text files you don’t want to clutter up with a licence statement), you can put it in the README or other documentation instead. Just be sure to clearly identify which files are under which licence. You can also put a licence statement into a file’s metadata, but that’s a bit more technical, both to do and to read—I wouldn’t count on users seeing a licence statement there.

What the statement says depends on the licence. For Creative Common licences, it can be pretty free-form; their licence chooser generates text like the first example below, but they have an “About CC Licenses” page that gives the second example.

• <Title of work> by <author> is licensed under a Creative Commons Attribution 4.0 International License
• © 2019. This work is licensed under a CC BY 4.0 license.

(Personally I’d go with something more like the first one, though I do like to use the abbreviations like “CC BY-SA” etc. I say that because part of the licence is that people using your work need to give you credit, so I think you should be clear about how you want to be credited.)

For the GNU GPL, they have a fairly specific guide on how to apply the GPL to your work. The licence statement basically has two parts:

1. An accurately dated copyright statement (e.g. “Copyright © 2022 Fan22”)
2. Three paragraphs that you can copy and paste from their page, starting with “This program is free software…”. (There’s an alternative, slightly longer version for works spanning multiple files, basically replacing “this program” with its name instead.)

As for the licence text, you can include a complete copy of the text with your work, as you described. The GPL recommends doing this, in a file called COPYING. Other licences are often found in a file called LICENSE. (If you go this route but have different licences for different parts, I recommend giving the files more descriptive names! Or I suppose you could put every licence’s text in one big file.)

Alternatively, you can link to somewhere that recipients can read it online—CC licences are usually done this way, with the link pointing to the Creative Commons website. (CC’s licence chooser generates the right link for you.)

A final note: Websites where you publish your work—including both GitHub and itch.io—often want you to list the licence on the work’s page. Itch.io lets you choose separate code and asset licences, which is great. As far as I can tell, GitHub will try and detect what licence you’re using based on a LICENCE or COPYING file, and it’ll only list one. I wouldn’t worry too much; the information you write in your files is still the authoritative word on what licence you’re granting people.