FBI agents approach Hannibal Lecter seeking his insight in tracking down killers but Lecter doesn't strike me as someone who'd help out of the goodness of his heart- what's in it for him?
Lecter doesn't strike me as someone who'd help out of the goodness of his heart- what's in it for him
Amusement and the chance to escape
In Silence of the Lambs, Lector isn't really trying to help the FBI, he doles out information, much of it false to amuse himself and perhaps improve his situation and, hopefully, escape.
When the information on Buffalo Bill is given to him he already KNOWS who the killer is and uses that knowledge to lead Clarice in one direction while extorting privileges from the Senator.
Once he is removed from his usual prison/hospital, he has abundant opportunities to escape, which he does.
One way to know this is that there is a CLEAR contrast to Hannibal's behavior in Silence of the Lambs vs his previous behavior in Red Dragon, where he seeks out Francis Dolorhyde and intentionally misleads former agent turned consultant Will Graham to almost deadly fate for incarcerating him.
In Red Dragon Hannibal is trying to get revenge. In Silence of Lambs Hannibal finds himself curious/attracted/in kinship with Clarice (which is also confirmed in Hannibal sequel film when Mason Verger is listening to Barney's tapes, hence why he uses her as bate), and he may still be trying to get back at the FBI internally by elevating Clarice's career, despite that he also has an agenda to escape.