I would have to say, "Yes!" It means that I am responding in the affirmitive, possibly in a grudging or reluctant manner. The source of the reluctance can be different, depending on the situation.
When is it used?
It is a way to soften the blow when the answer is "yes" but you don't want to hurt someone's feelings. "Does that hat look silly on you? Well, to be honest, I would have to say, 'Yes.'"
It is also used to provide some perceived distance from a position while still agreeing with the position. "Has President Bush exceeded his constitutional powers in this case? After reviewing the transcripts of the Senate hearings, I would have to say, 'Yes.'" Here, the implication is that the facts are what compel you to agree, not a preconceived notion or prejudice.
"Is this a valid phrase? In the absence of more context, I would have to say yes." Here, the reluctance is caused by a lack of information that is forcing a supposition.
I'm sure others will be able to explain more succinctly. I hope the examples help in some way.
It's an expression of tentative or noncommittal agreement.
Sometimes you ask a question that is a bit of a puzzler, or one that I haven't really considered before.
After a lot of consideration, I feels silly saying "yes, you're right," especially if it was a simple question-- all the old-timers here can tell you the brand-new learner can ask simple questions that prove most difficult to answer.
So, to account for the time I spend pondering over these simple questions, I will sometimes say "I would have to say yes!"
Meaning I'm still puzzled, but I've exhausted all the "no" possibilities I can think of. There's so much the native speakers of any language take for granted-- newcomers always notice, and they always want to know the why of it.
And of course we always try to answer. Problem is, we learned these expressions as naive children, and didn't question why things in our language are the way they are.
As for the grammar, it's very complex because it implies an "if clause" in the subjunctive.
"If I'm not mistaken," I would have to say yes.
"Since I can't come up with any reason not to," I would have to say yes.
"Unless there's something I'm overlooking" (due to ignorance I don't want to admit to)-- I would have to say yes.
Ignoring the exclamation point, I can think of two ways where this would be used. One would be that someone is asking a person a question where the person being asked the question is acting under regulation of a third party. e.g. if the person asking the question said "what would you have to do if ..." and the person being asked the question, knowing they are required by a third party (perhaps their boss) to say yes, would respond "I would have to say yes." The second and, I think, much more common usage of this would be when some is pretty confident the answer is yes but not completely sure, often in a hypothetical situation such as an ethical question or a result to an upcoming situation. e.g. A friend bought something at a store but it didn't work right and s/he still had the receipt and that person asked you if you thought the store would refund the money. You might respond with "I'm pretty sure they will" or you might say "I would have to say yes." There is definitely nothing grammatically incorrect with the first usage I mentioned, but I am not sure on the second.