Explain it like I'm 5! Seriously I've read people have extended the life of computers by installing Linux. What is it about Linux that makes it run better on older computers than Windows?
The short answer is 'less software bloat', partially due to having evolved from a design philosophy of being built up from small tools that do a simple job well working together, rather than swiss-army-knives that do everything. But mostly due to an absence of commercial pressures to do otherwise.
Much of a Linux environment is directly inherited from the Unix systems of the 1970s, when computers were a lot less powerful, so the basic operating system runs well on less powerful hardware. Obviously the 'desktop' applications (browsers, office suites, etc) are modern megaliths, but since you're not forced to have background bloat consuming resources, they can still run a bit better.
How to swap from Windows to Linux on an old PC?
You install a Linux distribution on the computer, in ostensibly the same way that you'd install Windows or MacOS on the computer.
Since you're asking the question, this probably isn't something you've ever done. In short, you'd boot from some sort of removable media (used to be a CD or DVD, but these days more commonly a USB flash drive) rather than the computer's hard disk (or SSD), into an environment designed for installing the OS. It will then walk you through erasing the contents of the computer hard disk, copying the new OS over, and setting it up to boot independently. You'd then reboot without the removable media, and the computer would be running the new OS.
(There's a lot that can potentially go wrong during this process, leaving you with a computer that doesn't have a bootable OS on it. It's less bad than it used to be, if only because these days you probably have access to another device so you can search the web for help.)
What are the benefits? Who does it suit most? Everyone?
There's no short answer to this. Stereotypically, it most suits people who know a lot about computers, because Linux environments tend to expose the user to more of the 'under-the-bonnet' stuff. Beyond that, its main advantage is that it's free, it works well on older hardware, and once you get everything set up the way you want it, if you don't meddle with things, it's generally more inclined to stay working without updates breaking or changing things.
My partner brought her old laptops home from work and there's 5 to 8 year old laptops that I'm wondering if Linux on them might make them OK for my 9 year old son to use. He's likely to just use for games and YouTube. Would Linux be a good idea?
Possibly. He might learn something.
Are there any games for Linux? Will the old laptop run at an OK speed? I don't suppose there's minecraft for Linux too?
IANAgamer, but:
Gaming tends to be where Linux lags behind
[1], because it's not in most people's commercial interest to specifically support it. I don't know much about gaming, but I believe some Steam games will work, and Minecraft is a possibility.
Performance will be in line with what you'd expect from the same computer running Windows. Games tends to be limited by the performance of the CPU and graphics hardware; a more efficient operating system is unlikely to gain you much (and poor support of advanced features of the graphics hardware can be a real probelm)
Finally how easy is it to install Linux on an old Windows laptop?
Somewhere between put-the-CD-in-click-'OK'-a-few-times-and-away-you-go and a circle of hell where you'll spend weeks banging your head against the foibles of BOISes and bootloaders and partition schemes and obscure device drivers, eventually get it to work, but discover it crashes whenever you close the lid or something. Basically just like Windows (except these days when you buy a computer, the manufacturer's already done all that shit for you, and sells you the thing that mostly works).
Unless you're doing this for learning's sake, I'd suggest you want to bring in someone with more experience. If you're lucky someone will have written about the specifics of installing a given Linux distribution
[2] on the specific model of laptop, which is always a good starting point, if only for determining that it's a terrible idea
[3].
The usual suspects will be along in a minute to tell you that:
a) Linux is for elitist nerdy schmucks, and don't wast your time on it.
2) You should use their favourite distribution, because reasons.
iii) Get a Mac.
[1] That and 'office' applications. You may have noticed that the world's leading vendor of office software is also a major OS vendor. You can probably work out why they don't release a version of their office applications for a free OS. There are some alternatives, which are usually fine for basic tasks, but fall short of being the Real Thing.
[2] The great thing about open source software is that anyone can come along, make a copy, change a few things, and release their alternative version. Different distributions tend to have a different focus. One might be designed to run well on old hardware. One might try to stay bang up-to-date with the latest media applications. One might be designed to be easy to use. One might be designed to be really secure. One might be deisgned to change as little as possible over time. One might be brown. Etc.
[3] Linux compatibility is a good litmus test for long-term viability of hardware. If it's well-supported in Linux, it'll probably stay well-supported in Windows or whatever too.