Is this real or my imagination?
 

I've notice when I'm riding, if I brace my arms slightly against the bars, whether on the tops, drops, or hoods, and push forward as if I'm gently pushing my bike by the handlbars, it seems to pick up speed.

I know this could have something to do with the force or leverage of my legs. Am I imagining things or is this an actual sprinting or climbing technique?

Relax your arms and lower your profile and without any more effort you will gain 1 1/2 to 2 MPH

Newton's 3rd law. When you push down on the pedals, the pedals push back on you. You're much heavier than the bike, so most of the force is used to propel you forward. When you grab the handlebars tight, you brace yourself against the frame to put down more force onto the pedals, and thus giving you more speed. That's why when you're sprinting, you're pulling on the bars.