In order to claim your next Content Pack you will need to visit the Twitch Prime Rocket League page after the drop date. The new Content Pack will show up the next time you start Rocket League after you have claimed it.
The second Twitch Prime Content Pack drops 27 days after your first claim. If you signed up on September 27th, the next drop is on October 24th. If you signed up after that, the drop will happen 27 days later.
The Twitch Prime Rocket League page will also tell you how many days until the next drop!
If you are eligible and haven't been able to claim your next Twitch Prime Pack, click the Submit a Ticket button in the upper right corner for help.