Why did Théoden suddenly trust Gandalf?

[Valandil]

I have a more traditional - or a more "tame" - take on it.

I think Theoden's "eyes were opened" so to speak. He realized quite quickly the road he had gone down, and the way he needed to get back.

Alcuin - you've caught up to me, I suppose. I'm in "Helm's Deep" right now.

[Olmer]

Originally Posted by Valandil

He realized quite quickly the road he had gone down, and the way he needed to get back.

Quickly indeed.
Picture this: an unwelcomed guest came in your house . He is demanding, threatening, meddling with everything you are doing, and literally getting so much on your nerves, that in a desperation to get rid of him you are offering him your prized car. Which he is taking and disappearing for good, as you thought. But a few months later (Surprise! Surprise!) he is coming back, this time more bold and bully. You already was fed up with his contrivances at the last time, so he is definitely not welcome. You did not want to deal with him and asked your trusted friend to tell him to hit the road. Instead he is giving to your friend such a blow, so he is turning from a wise man into a blabbing idiot. Then he comes to you and speaks in a "low and secret" voice, so none save you can hear what he said. Probably making an offer you cannot refuse. And then you have a dilemma: to kick his butt out, or to follow the offer. You don't want to wake up with a horse head in your bed, so you are getting up and do whatever he is telling you to do.
Only aforesaid scenario explains such a mercurial "eyes opening."
Or we would rather agree on the supernatural power of persuasion?