Although a variety of factors contributed to European success in American colonial attempts, animals were among the most important things that helped the Spanish, English, French, Dutch, and Portuguese establish viable colonies in the New World. It seems unlikely that they could have subdued the native populations they encountered and set up viable colonies without them.
Perhaps the most important way that animals helped early colonization was by providing a ready supply of food thousands of miles away from provisions. Livestock multiply, so with time, raising livestock is a way of increasing your food supply. They essentially convert that which humans cannot digest into that which they can. Not only are they good for meat, but they provide nourishing milk and leather.
Livestock are also good as beasts of burden. It is difficult to plow a field, but strong animal makes it easier. In an age when bio-power from the muscles of humans or animals is one of the only available sources of energy, animals play an extremely important function. They move what humans cannot. Given that humans are relatively weak, they need strong animals or some equivalent for most building projects or to conduct trade. Because the only beasts of burden in the Americas were Peruvian llamas, the Native Americans would have been hard pressed to build anything larger than the pyramids of Mexico City or conduct trade farther than a person could walk in a reasonable amount of time with only so many goods as he could carry on his back.
In an age when horseback is the fastest form of transportation, those with horses will have a great advantage over those who do not. Of course, horses are also a type of weapon. Since Native Americans had never seen horses before the Spanish showed up, they were afraid of them. The Spanish, on the other hand, could control when or if they wanted to fight because they could flee from a potential battlefield much faster than any of the natives could run.
Animals also carry disease. Since disease was a major factor in the European conquest of the New World, the importation of the rat to the new world is of vital significance. Certainly natives could and did pick up disease from Europeans themselves, but the importation of the rat essentially meant that it was too late for the Amerindians. Even if they had driven out the Spanish, they could not have driven out the rat and could have still suffered from plague and the resulting famine.
