A bud is a small, young flower, not fully formed.
To nip is a kind of pinching action with your thumb and index finger. (The finger you point with)
To nip something in the bud means to stop something in the early stages before it turns into a bigger problem.
Looking for a suitable clip, I was very tempted to go with the B52s - one of my favourite bands but sadly, nobody else's - it seems. So, here is something I didn't know existed, a comic preacher or maybe a preaching comedian. It's America, so I guess anything's possible.
Interest southern accent too. Praise the lord!!