The events in the Clone Wars novelisation by Karen Traviss are slightly different. Ahsoka doesn't speak Huttese but Anakin does. When Jabba threatens them Anakin is fully cognisant of what's being said and attempts to negotiate with his captor:
Jabba had his son back. He was barely able to believe it. Dooku had
conned him, but so had the Jedi. They were all the same, these humans,
only after his favor for what they could wring out of it in their
interminable little squabbles. He wouldn't let relief get in the way
of business yet. "Now, Jedi," Jabba said. "You still die." Anakin
decided he should have known better. It would take more than a tearful
reunion. if Hutts had that depth of feeling in them, to make Jabba see
"Okay, I'm the one you've got the problem with," Anakin said. "Let
Ahsoka leave with my astromech. She saved your son a dozen times since
we found him on Teth. She doesn't deserve this." Ahsoka's eyes darted
from face to face; she didn't speak Huttese.
It's pretty clear that in the Clone Wars film that whilst Jabba seems perfectly capable of understanding spoken Standard (it's used by his cousin on at least one occasion), Anakin doesn't understand Jabba, as evidenced by his lack of reaction to Jabba's accusations and pronouncement of their deaths. No explanation is offered within the film for this canon discrepancy.
Since we know that Anakin, a former slave of Gardulla the Hutt speaks Huttese, the only possible explanations are that Jabba is either speaking a dialect of Huttese that's unknown to Anakin or that he is speaking with an accent that makes it hard for Anakin to understand him. Either would explain why he's relying on the translator to understand what's being said.
Out of universe, it was simply a writer's mistake.