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.)

Actually, this isn't entirely true. You can follow the BOS story missions up to BUT NOT INCLUDING the mission "Tactical Thinking" without hurting your reputation with any other faction. If you follow their story line that far you will complete the mission "Blind Betrayal" which will remove Danse from the Prydwen (if you haven't already moved him) and will remove the threat of him being eliminated with the rest of the brotherhood if you decide to side with the institute. This mission also rewards you with a set of pretty good power armor so I find it's worth it to follow their story line through Blind Betrayal just to get it.