TBT is the forum currency - it actually is referred to just as "Bells" on the forum - but it's commonly called TBT by users. You earn it passively by making posts, and the amount you receive depends on the length on your post.
You pay by donating it to a user. There's a couple ways to do this...
1. You can click on the number of "Bells" you have in your sidebar, which immediately pops up a window to donate an amount to a user.
2. You can click your name at the top of any page, the select the Bells from there - this will take you to a page that has a Donate tab where you can give TBT to other forum members.
Those two places I've listed above also show you how much TBT you have. You have 33 as per your sidebar in that screenshot above.
On the forums, people use it to buy "collectibles" which can be found in the Shop (see second screenshot above - that last tab is the shop). You can also buy some quality of life stuff like username changes here for TBT.
Outside of forum specific stuff, they're useful to trade for in-game goods with other forum users. They're more consistent in value than in-game Bells or NMT because they can't be hacked or duped and you can't artificially give yourself more. So for example you could sell some wallpaper in-game for TBT, then use that TBT to buy some furniture in-game from another forum user. Obviously, the user has to be willing to accept TBT as a payment method.
Hope that helps, it's kind of a quick run-down I suppose.