Earlier quoted context omitted.
GNU Taler is a technology, not a company, so it doesn't "know" things in the sense that would be bothersome here. From GNU Taler's website: “When you pay with Taler, your identity does not have to be revealed to the merchant. The bank, government and exchange will also never learn how you spent your electronic money. However, you can prove that you paid in court if necessary.”
OK, the exchange will know your identity. That's not anonymous. Edit: Anonymity is a strong claim. Even Tor daemon warns "This is experimental software. Do not rely on it for strong anonymity."
A very simple example is using hashing. Let's say two friends A and B want to play head-or-tail in a chat discussion. If A first tells "head" or "tail" and then do the draw and win, B won't trust that A did not cheat. But if B do the draw, then A won't trust them. What you need is a way for either A or B to choose between "head" or "tail" and them commit to that choice and be able to prove it to the other after the other made the draw. This is possible without any trusted third-party:
— A choose for example "tail", they can hash that choice along with a "password" (generally called salt). For example they hash "tailLO7f-(86F" with md5, the result is "a079dcaae211e60756e5519058dcfc97".
— A sends that hash to B. B cannot know from the hash if A chose "head" or "tail", because the hash computation is way too difficult to inverse and they do not know the password to add after "head" or "tail".
— B draws a coin (or even chose the result, it doesn't really matter) and tells the result to A.
— Then A can prove to B that they chose "tail", by telling B the password "LO7f-(86F" so B can add that to "tail" and verify A's claim by computing the hash themselves.
Now this is a very simple example of how it is possible to prove things without disclosing information. It is of course much more complicated in the settings of financial transactions with multiple parties, but it shows that it is possible if necessary prove previously undisclosed claims.