One of the interesting bits of background about Spiegelman's text is that the mice were not originally drawn quite so cartoonishly. Instead, Spiegelman drew them in a manner that was more realistic/ human-like, clearly based in part on the famous photographs of concentration camp survivors taken by photographers such as Margaret Bourke-White.
Original sketch for Maus
Bourke-White photo from LIFE magazine of liberated prisoners
With this in mind, it's interesting to speculate about just why Spiegelman chose to revise his original mice into something more cartoonish. As Jennifer D. said in her response, does it have something to do with the icon and identification, as McCloud would have them? Is his decision to portray the Jews as mice problematic because it fundamentally essentializes race and trades in stereotype? Or, as we will see more clearly in Book II of Maus, is he trying to suggest something more subversive about race and national identity--and the combined ridiculousness and danger of racism? Does Spiegelman's choice to use animals in Maus encourage or forestall identification, in your opinion? Would you have responded differently to the mice if they were drawn in the manner of the more realistic, anthropomorphic sketch above?
Another important aspect of the Jews-as-mice, Germans-as-cats, Poles-as-pigs metaphor, as many of you have noted, is its historical accuracy in terms of the stereotypes in currency before and during the war, as well as the preoccupation that many Germans had with whether the Jews were a race.
Beginning during the eighteenth and nineteenth centuries, roughly coterminous with the Enlightenment's investment in creating scientific and racial taxonomies, anxiety about Jewish racial difference came to center around the Jewish body. Many Europeans became obsessed with the idea that Jews were members of an inferior, non-white race. Part of this obsession centered around the idea that Jews had distinguishable physical characteristics that rendered them decidedly other.
Sadly, many stereotypes found their way into caricature.
Above, you can see early examples of Jewish caricature that focus on supposed physical differences of Jews, including large noses and ears, oversized feet, and talon-like nails.
Later Nazi caricature traded in similar stereotypes of Jewish physical difference. (see below)
During this period of anti-Semitic propaganda, as some of you have noted, Jews were often likened to vermin, particularly mice and rats.
The Nazi film, Der Ewige Jude (The Eternal, or Wandering, Jew), one of the most disturbing anti-Semitic propaganda pieces, featured scenes that juxtaposed "typical" wandering Jew with images of rats and mice overrunning various parts of the globe. This image (a still of the rats can be seen below) suggested that, like vermin, Jews spread chaos and disease, literal and metaphorical, wherever they went.
For more on anti-Semitic stereotypes, you might check out resources such as Sander Gilman's work on The Jew's Body, this book on Racial Science in Hitler's New Europe, or this book on Nazi propaganda.
For more early sketches and background on the creation of Maus, take a look at Meta-Maus, Spiegelman's behind-the-scenes account of the creation of his work.