According to this article: https://developers.facebook.com/docs/messenger-platform/reference/buttons/game-play/ I can post parameter "payload" for button with "SERIALIZED_JSON_PAYLOAD" and "this data will be sent to the game". How I can receive this data inside my game?
You can use getEntryPointData inside the game in order to access the payload. Note that it must be valid JSON for this to work.