TL/DR: You can read about it in detail here: https://eprint.iacr.org/2019/1416.pdf
According to PrivacyGuides, regarding this announcement:
The payment uses the same zero-knowledge proofs as Signal's donation system, so they won't be able to correlate your payment info to you.
According to Signal's support article:
Is my payment information linked to my Signal account?
No. Your payment information is not associated with your Signal account. Using the anonymous credential scheme that we introduced for Signal private groups, clients make payments and then associate a badge to their profile such that the server can authenticate the client is in the set of people who made a payment, but doesn’t know specifically which payment it corresponds to.