Where @MichaelLorton's excellent answer appeals to the real-world history of the society that makes these stories, I've got an explanation driven by storytelling concerns.
When a conspiracy starts killing people, the audience immediately understands several things:
The conspiracy does exist
This becomes clear when people start dying unexpectedly, often when they are in perfect health.
The conspiracy perceives a danger to itself
Villains in the political conspiracy genre often get much less screen time than villains in other genres (and sometimes only at the very end of the film), which means they have limited opportunities to share their thinking with the audience. We don't get to see them tearing their hair and shouting angrily at subordinates to "fix this!"
The danger here is that viewers may not realize that the conspiracy is in trouble. A series of rapidly-executed assassinations helps establish that the conspiracy is sincerely worried, which helps us gauge how well the hero and law enforcement are doing.
The conspiracy is playing hardball
However great or small the conspiracy's goals may be -- from world domination to petty financial schemes -- if they're willing to murder people to keep it quiet, we know they are taking this very seriously. It proves they will go to any lengths to stay hidden, which helps justify whatever paranoid precautions the hero takes.
As audiences and writers have become more sophisticated, the genre has evolved to sometimes include really outrageous protective measures. As just one example: at the end of one such movie, we see Gene Hackman literally dismantle his entire apartment trying to figure out how the bad guys learned his secret. Our heroes would look a lot crazier to us, and thus less admirable and sympathetic, if it were not firmly established that misjudging the enemy and screwing up are guaranteed to be fatal mistakes.
The conspiracy is very capable
They are not amateurs, they are highly competent and driven. This becomes even clearer when we see that the conspiracy can accurately identify and locate witnesses who only got a glimpse of the crime, or who don't even realize they saw anything at all.
Finally, killing people who have forbidden information really does sound like an effective way to contain that information! (Do not try this at home.) As they say, "dead men tell no tales."
However, as we've all seen, it almost always backfires (in the movies). Here, I think the conspiracy genre often relies on assumptions we bring from detective fiction and (later) police procedurals: that it's impossible to get away with murder because there's always some evidence that will inevitably lead sufficiently smart and dogged investigators to the killer.
Of course, the killers don't believe that (because nobody would commit a crime if they were certain they'd be caught).
This makes it easy for the audience to accept that the conspirators are pursuing the course of action they sincerely believe is most likely to achieve a cover-up, while leaving open a path by which the conspiracy can credibly be uncovered.
The conspirators' assassination spree is a classic form of over-correction. Over-correction is a very natural kind of mistake.