How does Esri technical support work i.e. what kind of questions does it answer?

I became curious to know why people don't ask them to the technical support.

Many do also. When you have a issue or question, it can often be a good idea to request Tech Support (ie, ask Esri), and post to the ArcGIS discussion forum (ie, ask the Esri user community), and perhaps even also post here to GIS.SE. Doesn't have to be a this or that. Different perspectives, more ideas.

How the ESRI technical support works and what kind of questions does it answer?

Specific questions and issues are best. Think of it as a helpdesk or call center. You're stuck... help me now, type of thing. It's not a good resource to get trained from A-Z over the phone nor for more in-depth consulting. There are other services for that. But for specific problems, advice to get from one step to the next, helping finding more information or resources, troubleshooting, Esri Tech Support is really good. (disclaimer: I work for Esri)

I have contacted ESRI support several times in the last few years. You have to have your customer number, and alot of times end users in a large company do not have access to that number. My queries to them were usually about issues with ArcSDE that we HAD to figure out ASAP or extremely odd behaviors with ArcGIS Server. I find the email support to be quite good, and if you send them something in the morning, you should hear back by afternoon, otherwise it might be the next day; but always within 24 hours.

Also outside of the US we get local "Esri" support which is not always very good (sometimes never!). These forums allow us to tap into audiences in the US.