The main allure of this card is that it can be used to get out obscurely Ranked Xyz Monsters, like "Number 9: Dyson Sphere". However, it can be used in more stategic ways thanks to a certain card: Rescue Rabbit.

Since the early days of Yugioh had a large focus on Normal Monsters who now have disproportionate amount of attack points to the current game. Meaning that for any type or attribute, there is probably a Normal Monster version of it with less than 1000 ATK. And since Rescue Rabbit brings out 2 of the same monster, it makes a perfect opportunity to utilize Tannhauser Gate. (Ideally, you'd want to focus on monsters of levels 3 and 4, as doubling the levels of 1 and 2 would just get you 2 and 4, which you can already search out with Rabbit). This trick will let you get out the rarely used Rank 6 monsters. And what's more, it will let you get out the fearsome Thunder End Dragon without having to lose a high leveled Normal Monster.

Alternatively, you can utilize the effects of other monster who special summon themselves or others to set up a Tannhauser Gate. I've include a brief list here, sorted by type:

Spoiler:

Great job, Jake! Your "Rabbit Gate" strategy really pays off!