'I'm willing' means that I agree to do this. You may not want to do it, but you accept that it is necessary or useful. 'I'm more than willing' has the same meaning, but is more polite. If you say 'I'm more than willing to help', you are assuring the listener that you don't mind doing this at all. In neither case are you 'longing' to do this. Something that you are willing to do is never pleasurable, and never something that you have any desire to do. You may be willing to help someone, do some extra work, or pay for something, but you never long to do it. You're just agreeing or offering to do this.

You aren't terribly wrong but I think the nuance is more that I am willing and happy to do something. But not necessarily longing.

you shouldn't have said anything...my friend ususlly told me don't pause to do it

"More than willing" is emphatic or enthusiastic, particularly when a task might be considered unpleasant or burdensome. It has no implication of longing and is most naturally used with tasks imposed on you or requested by others. Being "more than willing" communicates amicability; it makes you pleasant; it makes you "a good guy." It can be used to put another person at ease; to tell them you are not merely willing but engaged; to let them know not to worry that you don't want to do it. It's also often used for contrast, paralleling something you will do with something you won't. "Sara, I'm more than willing to come over and help you paint, but I can't let you borrow my truck."