The events of human history on earth have revolved in great part around settling at or controlling strategically advantaged locations, for example any coastline, or a geographic bottleneck for trade and travel (think of Singapore and the Strait of Malacca).
A Lagrange point is the simplest space-based analog to this that I know of, if you want to put something in a fixed location relative to other bodies, the Lagrange points are places where you can do it with the highest fuel economy. Then when operating from that position you will have more energy available to do other things, granting you advantage over competitors who are not at the Lagrange point.
So whether it's science, research, trade, defense etc. there is a compelling reason to locate things at a Lagrange point, and it seems this is already happening as we have science satellites at L1 and L2 and I believe L3 has been talked about. The Lagrange points are not all created equal in terms of distance to their respective bodies, size, energy required to maintain a position etc. All two body systems have them, so for example the Earth and Moon have a set of Lagrange points that are significant to us.
The LPs are what a lot of our space politics and problems may eventually revolve around (quite literally!).