Any time I turn on either faucet (hot/cold) it starts leaking in the cabinet under. If I shine a light up to see where the water is coming from, it's a hole right in between where the handles are. The hole has a stick coming out of it which controls the stopper. That's where the water seems to be coming out from. Is there a easy DIY way of fixing this problem or do I need a plumber?

Remove the faucet handles. If there are any covers under the handles remove those. Turn on the water. You will probably see water coming out around the stems. Tighten the big nut around the stem about 1/8 of a turn.

I think HK is right the water is coming from under the handles themselves and should just need tightening or depending on brand, changing the rubber washers/o rings which are cheap and an easy to fix. You will know when you remove the handles, and crack the water open.