The culprit turns out to be a Rutan, an alien that (in its original form) looks a little like a cross between a giant jellyfish and a moldy lime.
The Rutans were attempting to wipe out humanity so they could use the Earth as their new home base, something that (if you believe classic Who) was the aim of every intelligent alien species in the galaxy and happened on a weekly basis, but for some reason this bunch of aliens decided the best place to launch their attack was a lighthouse out in the middle of nowhere. Be that as it may, by the time the Doctor and Leela foiled the Rutans' evil plot, all the people in the lighthouse were dead and/or vanished, so this definitely stands out as one of the Doctor's less successful ventures (although he did save the Earth, so there's that).
There are two curious things about this episode that are why it comes up today.
The first is that during its premier broadcast, on November 22, 1987, transmission was suddenly interrupted and replaced by a signal showing a guy wearing a Max Headroom mask babbling about random stuff (including his opinion of "New Coke" and the television series Clutch Cargo) and finally ending with him getting spanked on the bare ass with a flyswatter while a female voice shouted, "Bend over, bitch!"
The source of this transmission -- which I swear I am not making up -- was never identified.
The other strange thing about the episode is that it's based on a true story.
Well, not the green jellyfish alien part, but the mysterious deaths/disappearances from a lighthouse part. On December 15, 1900, the steamship Archtor was near the Flannan Isles in the seas off the Outer Hebrides and noticed that the lighthouse on Eilean Mòr, the largest island in the chain, was not working. They reported this to the authorities, but bad weather kept anyone from investigating until eleven days later.
When they got there, the lighthouse was abandoned, and the three crew -- James Ducat, Thomas Marshall, and Donald McArthur -- were all missing.
There were plenty of signs of recent habitation -- unmade beds, lamps cleaned and refilled, and so on -- but no indication of what might have happened to the crew. The lighthouse logs indicated nothing amiss other than some inclement weather, which is hardly unusual off the coast of Scotland in winter. It must be mentioned that there had been extensive storm damage downslope from the lighthouse; a metal storage box thirty meters above sea level had been broken open, presumably by the surf, its contents strewn, and an iron railing set in rock was bent nearly flat. Robert Muirhead, superintendent of the Northern Lighthouse Board, said some of the damage was "difficult to believe unless actually seen."
Still, it's presumed that the three missing men -- all highly experienced lighthouse operators, who had been on the job for years -- would have known better than to go out and walk the beach in the middle of a December storm. The lighthouse itself was undamaged, so whatever killed its keepers seems to have taken place outside the building. Muirhead's conclusion was that they'd gone out to try to secure the metal storage box that was later found damaged, and a rogue wave had swept them away.
There are two problems with this explanation. The first is that there was only one missing set of oilskins, implying that two of the men went out into a raging winter storm in their shirtsleeves. The second is that the worst of the damage seems to have happened after the lighthouse was abandoned; it was already not operating on the 15th, and the serious storms (the ones that prevented anyone from investigating for a week and a half) didn't start till the 16th. It's possible they were killed by rogue surf and/or bad weather, but this doesn't really answer all the questions.
So of course, this didn't satisfy most people, and that's when the wild speculation started. Sea serpents, an attack by the malevolent spirits of drowned sailors, abduction by foreign agents, and even that the three men had absconded so they could take up new lives elsewhere. A logbook surfaced claiming that there had been a devastating storm lasting four days -- from December 11 to December 14 -- bad enough that all three men had "spent hours praying" and Donald McArthur, an experienced lighthouse keeper, had "been reduced to helpless crying." The weirdest part about this bit is that contemporary weather records show no indications of an intense storm during that time -- as I mentioned, the seriously bad weather didn't really start until the 16th -- and certainly if there'd been a gale bad enough to trigger fits of weeping in a veteran seaman who was safely inside an extremely sturdy building on dry land at the time, someone on one of the nearby islands would have mentioned it.
However, the veracity of the entries has been called into question, and some investigators think the entire thing is a fake.
Then there's the fact that McArthur himself was said to be "volatile" and to have a bad temper, so another possibility is that there'd been a fight -- or perhaps a murder -- and after dumping the bodies into the ocean, the guilty party had thrown had thrown himself in as well out of remorse and guilt. However, there was no sign of any kind of altercation inside the lighthouse, and no notation in the (real) records left by the keepers that anyone had been acting out of the ordinary.
So we're left with a mystery. Three men in a remote lighthouse off the coast of Scotland vanished, and despite a thorough investigation at the time and a lot of speculation since then, no one has been able to figure out why.
Me, I'm voting for the Rutans.