It is true that "during constant acceleration, the average velocity for an interval is equal to the instantaneous velocity at half-time in the interval."

What I don't understand is, what is half-time and how do we obtain it?

Also, What does the area under a velocity-time graph give for a moving object? My guess is acceleration... is this right?

Thanks so much! :)

Constant acceleration is a straight line with a positive gradient on a velocity-time graph. If you make the 'average' velocity, you'll obtain the midpoint, i.e. where the time between the intervals is half (thus half-time). Half time can be calculated using the formula for average. Since you are using a time interval, you'll have only two time instants. Average (or half time) is:
(initial time+final time)/2

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Also, What does the area under a velocity-time graph give for a moving object? My guess is acceleration... is this right?

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Unfortunately, no. The answer is distance. You have the acceleration by the gradient of the graph.

Hope that helped!