What does “anti-addiction online time” mean?

Some countries have laws mandating how much you can play a day, aka anti addiction laws. The amount you see there is the length of your last play session. Some countries can use this to kick you off of the game to enforce the law. (Edited after reading some more info)

Why does a game shut down if the anti-addiction time reaches 1 hour?

Only if this anti-addiction time is a thing in the country you live in. It's not relevant otherwise, don't worry, game's not going to shut down. The counter itself basically tracks your total playtime for the latest day playing the game. If it exceeds one hour in China or wherever else this is active - game shuts down.

Is there psychiatric therapy for a young person addicted to Computer/Gaming?

In some countries there is even forceful psychiatric therapy on young people who are addicted to computers/gaming. If you are in the USoffackinA this does not apply to you obviously; so just open up another Red Bull and play the next match.