In the movie, the whole chain of events starts with Empire troops under command of Darth Vader attacking Leia's ship, Tantive IV, and taking her prisoner.
Since the Senate still exists at that point and apparently has to be reckoned with, it's a scandal ready to erupt. A Senator has been taken by force, some of her entourage killed, and no direct evidence of her involvement with the Rebels obtained. Other Senators won't like that (what if they are next?), and they have political influence.
So some subordinate of Vader brings it to his attention, and Vader orders to cover everything up (as seen in ANH), because he doesn't want any interference or delays:
Leia is marched away down the hallway and into the smoldering
hole blasted in the side of the ship. An Imperial Commander
turns to Vader.
Holding her is dangerous. If word of
this gets out, it could generate
sympathy for the Rebellion in the
I have traced the Rebel spies to
her. Now she is my only link to find
their secret base!
She'll die before she tells you anything.
Leave that to me. Send a distress
signal and then inform the senate
that all aboard were killed!
When Vader says that all aboard were killed, the Imperials either don't leave any living beings to tell otherwise or get Force choked. After such measures taken on Tantive IV, the landing party also has to cover up its actions. I know of no canon proof for this, but it's very likely that such orders have been given - why only hide half of the troublesome things? To anyone except few trusted by Vader, it must look like the whole skirmish has never happened. Which means, none of the people they find and question while trailing the escaped droids may talk to anyone about that. It's simply too strange to meet with stormtroopers in the middle of the Tatooine desert (where the Imperial military normally doesn't go at all), and the questions about some special droids with special data inside them don't help. Rumors can start spreading, and unwelcome questions can arise. If somebody comes looking for missing Alderaanian Princess, such a trail can be noticed and traced back.
And wouldn't it be so convenient if all those witnesses just died in an unfortunate accident involving some locals (well known to be aggressive), totally unrelated to any Empire representative, of course.
(This approach is unlikely to work on Mos Eisley scale, but stormtroopers are less strange when seen there, and they don't have to ask any detailed questions when looking for droids, so there's no need to eliminate witnesses anymore).
Without the initial events on the Tantive IV, there would unlikely be any need to cover up the murder of jawas. The stormtroopers could simply act as legitimate Imperial law enforcers. They could even kill jawas (as) if those resisted detaining and questioning, and nobody in the Empire would be held responsible.
Summary of TL;DR: because of the brute force use against Leia Organa, the political clout of the Senate could lead to something unwanted, so everything traceable to that has been covered up, with Vader's blessing, the witnesses have been killed, and those murders have also been covered up.