Why must the milk be frothed? Nearly every coffee machine has the milk go through this device which not only heats it up but causes it to froth up.

Why is this important for a coffee? What's wrong with just heated milk?
It's not that just heated milk is wrong or tastes bad.
It's only that there are some (popular) coffee drinks that uses steamed or frothed milk (are not the same, read this) by traditional way.

On those popular drinks you can find the cappuccino, latte, etc... (as you can see on this infographic), but at home you always can add regular hot milk to your coffee (but we all agree that it isn't feel and taste like the regular way).

Last, why are those techniques important for a coffee? I think that gives a different flavour and tasting to the drink, due to its consistency (and of course, that let you to do some pretty nice latte art)