What is a VPS and when do you need one?
A VPS is your own virtual server: guaranteed capacity and full control, from a few dollars a month. You need one when shared hosting starts pinching: running custom software, handling serious traffic, or hosting several sites. Self-managed means you need some technical skill.
On a VPS you get a walled-off share of a physical server, with its own operating system, memory, and CPU. Nobody can eat your capacity and you can install whatever you want: a specific database version, custom software, several sites side by side. DigitalOcean and Hetzner sell such servers from a few dollars a month, with Hetzner delivering remarkable power for the money.
The real price is responsibility: updates, firewall, security, and backups are your job. If you want VPS power without system administration, the middle path exists: Cloudways puts a management layer on top of cloud servers, so you get the muscle without being a sysadmin.
When is the moment? When your site structurally hits shared hosting's limits, when you want to run something beyond a standard site, or when you are a developer who simply wants the freedom. Until then, good shared or managed hosting is the wiser choice.