I tend to believe that the true usefulness of exploration doesn't become apparent until much later. Consider that the Europeans came to America to find India and access one set of resources... but ended up colonizing America instead. On the other hand, to justify any big project, you need something palatable to draw people's attention. One popular reason to build space infrastructure, besides the simple need for exploration, has been solar power. Such systems usually involve the construction of large solar arrays and microwave transmitters to beam the power to earth. I would like to propose an simpler alternative.
A non-negligable amount of the total electrical power is consumed by aluminum refining. In America, it takes up 1.6% of all electrical usage, of which about half is used up in smelting plants. It's pretty much an "energy bank". Nearly all of the original energy used to make the metal is expended during smelting, so by comparison, the re-melt comes at a minimum energy cost. The power requirements are such that aluminum refineries are located close to power plants with low rates per megawatt.
The ability to generate both power and heat from solar energy is well proven, both on earth and in space. There are power plants that produce hundreds of megawatts of power from the sun on earth and most satellites are solar powered. Solar power on Earth has many drawbacks. It is intermittent and somewhat unpredictable, being subject to reduced output from weather and also constrained by daylight hours. Furthermore, the atmosphere absorbs some of that energy. In space, it has 99% availability and no atmosphere in the way. Furthermore, structural requirements are relaxed such that lightweight components like inflatable mirrors can be used.
Thus, given that the composition of the moon's crust contains as much aluminum, titanium, and silicon as earth's crust, refining lunar aluminum, titanium, and silicon in orbit can create a space industry that has many of the same benefits as solar power while requiring less technological development. This has the advantage of re-using the infrastructure necessary to make the satellites in the first place, producing higher quality materials, and offering easy bulk transportation.
First, in terms of building a large enough solar power plant, most studies have concluded that it's better to use resources available in space instead of shipping them from Earth. Furthermore, both mining asteroids and the moon allow us to use technologies that might not be acceptable to deploy on Earth. For example, we can build mass drivers on the moon or push asteroids into Earth orbit with nuclear-powered engines for raw materials.
Second, the materials processed in space can be at a higher quality.
On Earth, in order to make aerospace grade materials, a vacuum melt process is often used. This allows lighter elements contaminating the bulk metal to "boil off". Often times, several melting cycles are used. This is especially apparent in Titanium production. Given that space is already at a better vacuum than Earth and this vacuum is essentially "free", in the process of being fabricated, all ingots will be subjected to at least one vacuum melt.
Similarly, both in the semiconductor and aerospace industry, single-crystal materials are necessary. For semiconductor fabrication, each crystalline defect reduces yield and increases cost. For aerospace components, crystalline defects reduce the structural strength of the part. Results from the Wake Shield Facility and other space experiments show that space platforms can more easily create giant defect-free crystals.
Finally, consider transportation. Transporting materials up from the ground is expensive. Returning bulk materials to the ground is inexpensive. Perhaps as simple as creating a giant ingot and landing it uninhabited land, covering it with sufficient slag such that it survives the trip through the atmosphere without appreciable oxidation of the ingot.
The effects could be huge. This will reduce the demand of aluminum manufacturing on the power grids that is increasingly being fed by coal powered refineries in China. It also can reduce the cost of aerospace-grade materials allowing things like titanium or aluminum car structures. Finally, because it has the potential to reduce the cost of single crystal silicon wafers, it also makes solar power on earth more feasible.
Because many space colony and space power plant plans require first the creation of a space manufacturing facility and then the creation of the project they target, this provides a valuable and feasible way to cut that early cost and allow the first stage of the project to start producing some income earlier.