William Porter's answer is largely correct, but incomplete.
Haste grants an additional attack only on Full Attacks, not on the Attack Action (which is a Standard Action).
The Staggered condition restricts you to either a Standard Action or Move Action on your turn, and since that makes Full-Round Actions impossible you cannot use the Full Attack Action to make a Full Attack.
However, if you have other ways of making a Full Attack without using the Full Attack Action, you can still benefit from Haste and use your additional attack.
For example, while Staggered, you are allowed to perform a Charge attack with only a Standard Action at a reduced movement. If you have the Pounce ability or something similar, you can still perform a Full Attack at the end of the Charge and then use the additional attack from the Haste spell.
Paizo released some FAQ regarding this: