First of all, even though you probably don't want to hear this, there wouldn't be an equally interesting and entertaining movie without the humans. While an experiemental movie around animals would be otherwise interesting, it would certainly not have the mainstream action thriller appeal together with its emotional drag and the philosophical implications for our own understanding of reality. But so much to the obvious out-of-universe answer to the question.
Given this we might still speculate about a possible in-universe explanation and there are multiple possibilities, even if a defite answer is most probably lacking. Let's first accept the premise of the movie, that the machines need livestock to produce energy (even though that's debatable in reality) and that humans need a matrix in order to keep entertained and not degenerate.
I might second CrowTRobot's answer that you would maybe need carnivores because of the lack of plants. But even then you could use other carnivorous animals. But if we accept the premise that humans need a matrix for mental occupation, then what tells you animals wouldn't need one, too? Ok, this matrix might be simpler than 1999's human society, but I'm not sure that's such a big problem for the machines at all.
There might very well not be any animals left on earth, either because humans reached a stage of civilization where animals have become nearly extinct, or they were killed in the crossfire of the, certainly world spanning and devastating, machine war, or they just degenerated in the wake of the nuclear winter (after all the humans only survived with the help of the machines either).
It is in a way more efficient in that you on the one hand have your enemies out of the way and on the other hand can use them for something meaningful, instead of just piling their corpses. With animals for energy production the "human question" is still not solved. So why not use the humans for energy production instead of killing them and trying to find other animals as batteries.
Last but not least, energy production might not have been the only reason the machines kept the humans for and this might be grounded in the humans' special status in the eyes of the machines. As can be seen in this related question there are various other theories for the machines' interest in farming humans, like punishment, preservation of the human race, or harvesting other things apart from plain energy.