I'm going to try and answer this as clearly as I can as I have just worked out how to get everything working. This is how you connect power to objects in your settlement:

• What you want to do first, as I realise you have already, is get the source of power, i.e. your generator
• Next, couple the generator with one of the 'power pylons' from the 'connectors and switches tab'. Add a cable between the generator and the pylon

• The next step is optional; add a 'switch', again from the 'connectors and switches' tab, and place it on a wall somewhere. Attach a wire between the pylon and the switch

  Some lights, like the 'construction' lights, can be wired straight into the switch, and when you come out of workshop mode, you just flick the switch and they will come on

• For other lights, TVs, etc you will need a 'power conduit', again from 'connectors and switches'. There are three types of conduit, there is one for ceiling, wall and floor, and depending on which of these surfaces your desired item is going on, you choose the appropriate conduit. e.g. if you want a wall mounted light, you add a wall mounted conduit

• Wire your conduit to the switch or to your power source that you installed earlier, then, add lights and TVs to your hearts content. You don't need to wire them to anything

You must have copper in your (or your workshop's) inventory to attach a wire, although you can delete a previously-built wire for a refund of copper used.

After some experimentation, and expanding on information in the comments:

Short answer

1. Because of bug.
2. But it doesn't really matter.

Long answer

[The problem]

This happens consistently in my Sanctuary settlement too, the only settlement where I have spotlights installed. I have two spotlight turrets; each is on level ground, and each is right next to its own dedicated generator.

The problem appears to be triggered by the passage of time rather than fast travel itself. For example, a very short fast travel (e.g., to Vault 111 and back) leaves the spotlights on, but a little further (e.g., to Starlight Drive In and back) and they'll turn off.

But most importantly, the problem is only cosmetic.

[The experiment]

Wait until a spotlight turret is in its darkened state, then:

1. Note that it's still moving; this strongly suggests that the game still considers it to be active.
2. Go into Workshop mode and notice your settlement's defense number.
3. Pick the turret up and set it back down. It should be lit now.
4. Notice your settlement's defense number is the same whether or not the turret is lit, so long as it's moving.

Then, to be extra sure:

1. Fast travel a good distance, for example, from Sanctuary to Starlight Drive In.
2. Return to your base. The spotlight should be dark.
3. Check your settlement's defense number. It's the same.

[So...]

While a little lighting in your settlement would be convenient—not to mention darned pretty—this appears to be a purely cosmetic bug.

You can sleep soundly knowing that those turrets, even when dark, are scaring away bad guys. And you can also sleep soundly because those turrets won't keep shining in your window. (Until Bethesda fixes this bug.)