I'll try to give some high-level intuition of the used construction paradigms in my answer below. First, note that we distinguish between one-show and multi-show anonymous credential schemes.
For one-show credential schemes, you essentially have that multiple showings of the same credential can be linked to each other, while showings are unlinkable to the issuing. Those schemes are typically instantiated using Blind signatures (e.g. Brands' credential scheme, or Baldimtsi and Lysanskaya's anonymous credentials light). Essentially, a Blind signature scheme is an interactive protocol where the signer does not learn the signed message. The issuer usually signs a blinded version of a commitment to the attributes of the user and the user then uses the unblinded, signed commitment together with a proof of knowledge upon showing. From a very high-level point of view, the property that the signer does not learn the signed message helps to achieve unlinkability between issuing and showing.
For multi-show credential schemes you essentially have that even multiple showings of the same credential are unlinkable. Those schemes are typically instantiated using a signature scheme where the signatures can be publicly randomised and which is compatible with zero-knowledge proofs. The issuer signs commitments to the attributes and upon showing one randomizes the signature and proves knowledge of the signature and the attributes (only the attributes which should be shown get revealed). The most prominent instantiations of this paradigm are Camenisch and Lysyanskaya's anonymous credential schemes (both the RSA variant and the pairing variant). Recently, another paradigm to construct multi-show anonymous credentials was introduced by Hanser and Slamanig. The use signature schemes where signatures and signed messages can be publicly randomised, together with a compatible commitment (i.e., one which still binds the committed message even when re-randomised). Here, the issuer issues a signature on commitments and showing amounts to re-randomizing the signature and the signed message and opening the commitment with respect to the attributes to be shown (they also require a simple zero-knowledge proof to guarantee freshness).