The story has become an internet sensation over the years, with many people blaming aliens, and then ghosts, and then the yeti, or possibly all of them working in tandem.
"So we're agreed then: We tear up their tents, take a lady's tongue, and never tell a soul."
The Obvious Answer:
So there's six things that freak people out about this one:
1. The no-tongued woman
2. A mysterious orange tan on the dead bodies
3. The ripped tents
4. The hikers' lack of clothing
5. The crushing damage done to three of the hikers
6. The traces of radioactivity
The big fact that gets lost in the re-telling of this story is that the bodies weren't found until weeks later. It's not like somebody turned their back, then five minutes later all their friends were dead and half naked.
That makes the missing tongue a lot easier to explain. As disturbing as it may be, the first thing a scavenging animal is going to go for is probably the soft tissue of an open mouth, especially if it still smelled like the burrito the hiker just ate. Laying out in the sun surrounded by white snow for days also accounts for the weird tan.
The trauma and the destroyed tent points to an avalanche. Their state of undress can be explained by paradoxical undressing, a known behavior of hypothermia victims when their brains start to freeze and malfunction. In other words, it's the kind of behavior you'd expect from a group of injured avalanche victims wandering around in the middle of the night in the freezing cold.
What about the radioactivity? Or stranger details that turn up in some accounts, like orange lights in the sky? Well, there's the fact that none of that stuff turns up in the original documents from the incident, and appears to have been added later by people who just can't resist making things spookier than they are.
It's those later accounts that have stuck in the public memory, because so many of the original reports were destroyed (this was the Cold War-era Soviet Union, which treated casserole recipes as state secrets).