Aside from human error there is no plausible explanation made explicit in the book except for plot convenience and lazy writing.
There’s no fun in that so I have conjured up a relatively plausible albeit purely speculative explanation that is at least backed up by some evidence.
<PERSON> is a relatively powerful and talented wizard (e.g. he learned how to transform into a rat while he was still a student at Hogwarts), as well as a very sly individual who has been hiding out and looking over his shoulder for more than a decade. There are also numerous examples of spells and potions that can confuse or even control people, including the Confundus Charm (which is mentioned soon afterwards in POA), the Befuddlement Draught, or even the Imperius Curse.
It is therefore plausible <PERSON> uses magic to confuse people and to coerce them into making decisions that benefit him. Whether he cast some kind of charm long before the climax of POA or in the moment (wizards can perform magic even without wands, though maybe not as strongly), clearly it is otherwise impossible that between the three teenagers and two adults in that room not one of them thought chaining a dangerous and slippery animagi to an injured boy would be a good idea.
On top of that you have <PERSON>, who after decades of being careful suddenly forgets about being a werewolf and messes everything up and almost gets everyone killed.
<PERSON>’s protective charm isn’t strong enough to confuse people enough to let him go outright but it can cloud their judgement enough to provide him with an opportunity for escape due to poor decision making.
It helps that no one was in a particularly clear state of mind: * <PERSON> was seriously injured and shocked at the revelation that his pet was a man all along. 194
* <PERSON> quickly went from thinking his parents were betrayed by <PERSON> to realizing it was <PERSON> * <PERSON> has worn herself thin taking too many classes all year and messing with time travel * <PERSON> is about to turn into a werewolf plus he realizes he’s been wrong about his friends for years * <PERSON> has been in Azkaban for twelve years and barely managed to hold on to his sanity. Plus he hadn’t planned on keeping <PERSON> alive at all anyway.
The three teenagers are also inclined to follow the lead of the two older, more experienced wizards.
<PERSON> used a similar technique to convince <PERSON> to make him the secret keeper for the <PERSON>. He used it again when he escaped from <PERSON> for the first time and faked his death. Maybe he didn’t have to use magic for those tricks but either way it is evidence of his ability to manipulate people and escape from sticky situations. 725