The Wild Thornberrys/WMG

Nigel Thornberry is a Time Lord
His strange behavior is because he's unfamiliar with the customs of 20th-century Earth, and he got a job hosting a nature show because he was curious about the "exotic" wildlife of Earth's distant past--naturally, the Commvee is a disguised TARDIS. He probably secretly had a hand in Eliza getting her animal-talking powers through some kind of sufficiently advanced technology, because he knew that it would make the job a hell of a lot more interesting.

Eliza awakened her Jacobson Organ just like in Regenesis
In the Cannadian TV series Regenesis one of the protagonists, Dr. Bob Melnikov undergoes an optic nerve reconstruction using a virus and awakenned his Jacobson Organ that gives him the ability to "feel" the other people feelings through the Pherormones. But... in real life, the Jacobson organ is important in the social aspect of animals and their way of comunicate. So... the Shaman or someone else reactivated Eliza's Jacobson Organ in some way and then she gain the ability to talk with animals: She feels the pherormones and her brain "translates" this smells to words.
 * But in Regenesis... Dr. Melnikov awakenned his Jacobson Organ with an HSV-1 virus... so that mean that the Shaman or any other, awakened her Jacobson Organ in a very disturbing way.

Debbie will grow up to be an anthropologist of some sort
She finds animals boring, but as much as she complains about the world trip she really quickly makes human friends and often adapts well to other human cultures. I honestly think she may become an anthropologist or something similar, maybe even a documentary maker like her parents but about other people and cultures and not animals.
 * Kudos for mentioning anthropology. This profession Needs More Love!

Eliza is just pretending she can talk to animals to cope with loneliness.
Eliza is pretending to talk to animals, in a way similar to having an imaginary friend. She doesn't get to talk to many kids her age, so it's just a way to deal with her family's lifestyle. Notice how she gets really excited to be around any kid her age, and how the animals she talks to tend to deal with the same problems she is dealing with.