In this context, it means "I figured it out." or "I now understand the thing that was confusing before." It is almost like saying "I solved the mystery."

I think that "I worked that out" is a perfectly acceptable way, when speaking informally, for native speakers of American English, at least, to tell someone, "Yeah, I figured that out." We all have a continuum of levels of language that we feel comfortable using in different situations. What we have here is, of course, a back-and-forth conversation between a woman and a man who are on a date. The woman says "I just thought you ought to know (that I have a child)." He asks her how it makes a difference. "The man indicates that for a fleeting moment he didn't fully understand what she meant, but then it seemed obvious that she is talking about how a man may react when a woman he is romantically interested in reveals that she has a child. A date can mean merely that two people are doing some enjoyable activity together as friends, when neither of them want it to lead to a romantic relationship. For example, when I was a law clerk, a lawyer at the firm where I worked asked me to go with her to see the movie "Far and Away" and I did. But we were just friendly workmates, nothing more. In years past, the word "date" almost always implied that the man was a suitor courting the woman (though it may now be the other way around). So a "date date" would be that traditional type of courting, not just doing something with a friend.

It sounds like one person has revealed that they have children. So the person that is being told says "I worked that out" -- meaning they went through a process to understand this, but it's really not a great way to say this. Most people would say, "I figured that [out]." When you 'work something out' it usually means you are working to solve an issue. That's why in this case -- in my opinion -- it seems a bit inappropriate. Second, a "date date" is a little bit of funny language we use when we are making an adjective out of a noun to emphasize it. What they mean is: This isn't just a coffee date or something platonic. This implied that the time spent together is to enjoy each other's company in a possible romantic way. I think the reason we use this double-word emphasis on "date" is because back in the 1900s, when you said you were going on a date, it meant a romantic date and it was pretty much understood. Now we have "play-dates" for kids, "dinner-dates" that don't involve romance, we "set a date" when we arrange meetings and appointments. So the word "date" sort of lost its charm, especially in recent decades. So if someone asks, "How about we go out on Friday?" the other person might ask back, "You mean, a date-date?" Depending on their tone of voice and reaction, it can allow the person asking the other one out to imply their intentions. "No, I don't mean a date-date, I mean just dinner, as friends." Or: "Yeah, silly, I'm asking you if you want to go on a date-date." :)