Nolo.com provides a basic overview of how pet trusts work here. Essentially, they are no different in concept than a regular trust or will. The idea is that pet owners care deeply about their animals, and want to ensure their continued care should the owner pass away. The case of Leona Helmsley underscores the importance of pet trusts.
The book, "Litigating Animal Disputes" provides a definition of the pet trust on p. 91
"A pet trust is a legal technique an owner may use to ensure the pet receives proper care after the owner dies. The pet owner creates the trust either in his or her will (a testamentary trust) or in a trust document (inter vivos trust). By the terms of either trust, the pet owner gives his or her pets and enough money for the care of the pets to a trustee who is under a duty to make arrangements for the proper care of the pets according to the standards outlined in the trust."
This tab provides helpful secondary sources regarding pet trusts, as well as sample documents and practice guides for attorneys who may encounter the scenario in practice.