I (15) am not allowed to smoke in Germany. Why not? What is the Jugendschutzgesetz (youth protection act)?

The youth protection act is intended to protect children and young people from harm.
It applies in particular in public, meaning outside your home.
Adults may not allow children and young people to do some things here.
If they do, the adult can be given a penalty.

The law differentiates between children and young people.

What is forbidden for you?
You may not be in restaurants or pubs after 11 pm.
You may not play gambling games for money.
You may not drink alcohol and you may not smoke.
You may not buy or rent films which are not ‘approved’.

Each film is assigned an age rating. If you are younger than this rating you may not watch the film.
This is on the packaging.
These rules also apply to games consoles, computer and mobile phone games.

Take a look here to find out more:
For adults: you can order the law in different languages for young people here: