It's impossible to answer for all people, obviously, but my students find 20th century history most relevant. It's an explanation of how the world that we live in directly got to be the way it is. Obviously history is continuous — the 14th century is also part of the explanation of how the world got to be the way it is — but the more recent the history, the more seemingly apparent the relevancy. Separately, because the 20th century is so proximate, it does not require nearly as much "work" to make sense of. Trying to make sense of the 14th century is non-trivial, to use my arbitrary example: it's so vastly different from the present that it truly does feel like a "foreign country." Making sense of, say, the fear of Catharism, requires a lot more background work than does making sense of, say, the fear of Communism. And even the latter is becoming difficult for students born in the 21st century to understand!

World War II is of particular interest, I believe, for a few reasons. One is that it did set up much of the "modern world" in its aftermath. The other is that it has been so valorized and riffed-on in our cultural legacy that it is in some ways quite familiar. It has been turned into a simplistic "good versus evil" conflict on a grand scale, in a way that both earlier and later conflicts have not been (they tend to be depicted as shades of gray). It has become part of our cultural archetypes (at least in the West) for thinking about just war, the price of struggle, and the depth of human depravity. So it resonates in a way that, say, the Korean War does not.

Lastly, I would note that in my experience when it comes to "what are laymen interested in?" (as someone who does a lot of scholarship for laymen), part of the answer is: "they like things that extend knowledge they already have." Which is to say, it's a lot easier to say, "you already know about N, but now, here's N+1!" As opposed to saying, "you know about N, but here's X, something totally different!" The latter takes a lot more work to assimilate into one's mind, and that's why you see so many new history students (on here and elsewhere) asking how on earth historians hold so much information in their head (it's because when you start, you are starting from scratch; as you persist, it becomes an N+1 situation, which is very easy to adapt to). So the most publicly potent things I write are the ones that take a character or event that a large number of people already have associations with (World War II, the atomic bombings, Richard Feynman and Albert Einstein, etc.) and then contrive to add a new spin or take on them (which adds novelty). "You thought it was this way, but it turns out it was really this other way" is the most potent way to grab eyes a historian can do (and is not without its controversy, especially if you are — in the eyes of your audience — taking something cherished and showing it has a dark underbelly).

All of these things naturally, I would theorize, culminate in exactly the sort of phenomena you are describing: that when laymen become interested in history it is typically in the history of things they already know something about, and look like they have obvious relevance to their present-day lives and world. Of course, there are plenty of other reasons to be interested in history, and one also finds those reflected to — some people prefer the almost escapism of very-long-ago history, which serves a somewhat different social function, I think.