Why is the Diels-Alder reaction classified as a 4+2 cycloaddition?
Because it generates a new ring ("cycloaddition") and it involves the participation of atoms that span four bonds of the conjugated diene and two bonds of the dienophile.
The mechanism is as follows:
where the conjugated diene must be s-cis for a reaction to occur. Otherwise, carbons 1 and 4 on the diene are too far apart to react in a concerted fashion. (Hence, trans-1,3-butadiene would not react.)
And they're pretty much all like that (with heat), when the reactants are symmetrical or don't have any electron-withdrawing/donating groups.
You can see that with carbons 1 and 4 on the conjugated diene and carbons 1 and 2 on the dienophile, as well as the generation of a new ring, we do indeed have what constitutes a [4+2]-cycloaddition.
You might want to refer to this answer for more information on the Diels-Alder reaction.