About self-hosting
Self-hosting means owning and administrating your own server, typically at home, to host your personal data and services yourself instead of relying exclusively on third parties. For instance, you can self-host your blog, such that it 'lives' on a machine that you have control of, instead of having it on somebody else's computer (a.k.a. FAANG or The Cloud) in exchange for money, advertisements, or private data.
Self-hosting implies owning a server. A server is a computer that is typically accessible on the network 24/7, and usually does not have a screen or keyboard (it is instead controlled remotely). Contrary to a popular belief, a server is not necessarily a huge and extra-powerful machine: nowadays, a small, ~$30 ARM board is adequate for self-hosting.
Self-hosting is not about making "your Internet" more secure and does not provide anonymity by itself. Instead, it is about being autonomous and in control of your services and data — which also means being responsible for them.