There are two fundamentally important facts that make carbon fibre a “good” material:
1) High Specific strength. Not the use of the word “specific”. In this case, this indicates that carbon fibre is strong for its mass. This is important, in that you can make a strong bike out of literally anything, but that it will be lighter and smaller when made out of carbon fibre.
2) It is a material that is laminated form layers, and as such, the finished part can have a different strength and a different stiffness in any given axis. Unlike say a lump of metal, which unless specifically machined, has the same strength in any direction. This means, that when properly designed and manufactured, the strength can be utilised in the most useful direction, rather than “wasted” in directions that do not carry loads etc
Now, the down side:
3) the fact it is an engineered and laminated material means the strength of the finished material depends highly on the way it is manufactured. It is easy to accidentally get voids in the material, or the resin to fibre ratio wrong, or lots of other factors, and as such, it’s easy to make parts that look strong but actually aren’t.
Plenty of people have made “homemade” carbon bikes, but i suspect that without access to the proper design methods (including fibre loading simulation) and with limited tooling budgets and manufacturing capability (ie autoclaves, laser cutters etc) there frames are going to be pretty much as heavy as an alluminium frame, because they cannot ustilise the absolute strength of the material.