Here's why:
You see a bunch of day laborers waiting outside of Home Depot awaiting hire.
You turn to your buddy and say, "freakin' illegals, they took our jobs". Your buddy turns to you and says, "greasy wetback Mexicans". In this context, what you said, and what your buddy said, are equally heinous statements. Why? They may be perfectly legal immigrants, you have no way of knowing. They may or may not be from Mexico. The could be Ecuadorian, Guatemalan, Honduran, Salvadoran, etc., and so on.
In other words, "illegals" is used as an epithet similar to "paddy", "kraut", or "***", with "***" (without papers) being the most apt and equivalent. If you want to refer to illegal immigrants, then use illegal immigrants.
In true meaning, illegals is dehumanizing. The implication is that the person to whom you are referring is illegal, illegitimate, invalid, not of the same worth as one who is not "illegal". But in fact, it is not the person who is illegal, but the circumstance of that person's travel across the border. This alone, regardless of bias, should clearly illustrate that the term "illegals" is nothing but a slur.