My question is the title itself: what is homebrew (or home brewing)? And how do homebrews differ from ordinary ham radios?
Homebrew is (as the name suggests) when you make your equipment yourself, as opposed to buying it.
An amateur setup consists of many components - the largest and most complicated of which are a radio transceiver (or separate receiver and transmitter), and the antenna. A power supply is also needed, but these are relatively inexpensive compared to the cost of other components.
Many people use a homebrew antenna, as it’s relatively straightforward to make a simple dipole and all it really costs is some wire and a balun. However, building a transceiver is a much more daunting task, unless you’re happy to stick with a single band and QRP.
Some people became radio amateurs literally because they wanted the challenge of building large complex electronics projects that they could maintain and improve, while using the equipment on a daily basis. However, for us mere mortals it is beyond most people’s ability without a long time of research and learning, when you can spend money and get a reliable working radio.