You can change a bot's description (as well as its name, picture and other details) only through @BotFather.
The bot description is a short text that describes the bot. Users see it at the very start, under the "What can this bot do?" heading.
To change the bot's description:
1. Open the @BotFather bot.
2. Send the /mybots command to @BotFather.
3. On the keyboard below the message, select the username of the bot you need.
4. On the keyboard below the message, select "Edit Bot".
5. On the keyboard below the message, select "Edit Description", enter the new bot description and send the message.
Done! The bot description is set.
Important! If the description has not changed, wait a few minutes or restart your Telegram client.