Attack speed is the number of projectiles fired from your tower.
For very low attack speeds, the attack speed value is equivalent to projectiles fired per second (before factoring in Rapid Fire, Multishot, and Bounce Shot). For medium to high attack speeds the attack speed value may be closer to bullets fired per 2 seconds and for very high attack speeds (over 40) the amount contributed per point in attack speed is further diminished.
You unlock the Attack Speed lab in Milestones by unlocked labs, which you do by reaching Tier 1 wave 30.
Attack Speed - Lab values, costs, and duration (99 levels)
The Attack Speed Card is a common card that increase tower attack speed by [X].