To get the Captured King Tiger you need to:
Be a Twitch Prime member (i.e. have Twitch Prime subscription).
Link both your Wargaming.net and Twitch accounts in the Wargaming.net Account Management Page. If you’ve done this already, you don’t need to do it again.
Claim your Captured King Tiger on the offer page*.
* For the tank to be credited to your account, you will need to launch the game after claiming it. If you are already in the game, please close it and log back in.
If you claimed the tank once, there’s no need to re-claim it every month when a new content package gets revealed.
If you discontinue your Twitch Prime membership, the tank will become unavailable for battle, but will not be removed from your Garage.
If you unlink your Twitch and Wargaming.net accounts, the tank together with the Slot will be removed from your account. The crew and 3D style will be sent to the Barracks and the Storage respectively.
All other content items from Care Packages (to be claimed separately) remain. To get the combat-ready tank back, you’ll have to become a Twitch Prime member again. The tank will be granted back to you, but you’ll have to manually recruit the Crew and install the 3D style.