In the Harry Potter and Sorcerers Stone movie, why does Snape give Harry that look when he states they are "up to something'

Since the query specifies the movie, here's the scene in question:

Up to Something!

Real world answer: this is just actor Alan Rickman being totally awesome, in character, and on point.

If you watch the clip, Snape looks curiously at Harry twice. When he's saying "three young Griffindors such as yourselves", his expression is one of mild suspicion, as any teacher would have when three troublemakers are plotting in the halls. And he continues to stare at Harry for several seconds -- at least five seconds.

He then switches the same mild suspicion gaze over to Hermione, who begins to stutter an explanation. As she's trolling about for something to say, Snape's expression turns to a fully arched eyebrows "this is going to be good" sort of expression. He's clearly enjoying her discomfiture.

When he tires of pulling her on the hook and begins to say "you want to be careful", his expression changes to one of his ordinary disapproval of all things Griffindor.

Notice that right after he says "careful", his body kind of jerks just slightly, perhaps as if he's sensed something. Perhaps some thaumic vibration, some random thought.

But, when he goes on to say, after one of those glorious Snape pauses, "up to something", Harry gives him a very determined stare, a challenging stare, and there follows a very clear change in Snape's entire demeanour: his eyebrows lower, his eyes narrow, his glabellar furrow deepens and his whole expression intensifies far beyond what's needed to scare a couple first year students.

He appears genuinely concerned about something. Perhaps he has discovered something via legillimency. After he finishes speaking, his expression turns somewhat thoughtful, as if he's discovered or deduced something of great interest that needs more consideration.

As Snape turns away, his eyes continue to be focused on Harry.

Conclusion:
Where Severus Snape is concerned, every word, every action, every slightest inflection of voice or expression is laden with deep layers of meaning. Perhaps far beyond that of almost any other character.

From hindsight, we know more about Snape and his curious relationship with Harry: the movie was released in 2001, and we won't even be able to read the final book until 2007. I stand by this as being one of many examples of the actor absorbing his character, even though no one yet knows the full implications of the story and presciently revealing that character admirably.

So, yes, the look definitely means something! What it's full implications may be are not fully revealed until much later.

The scene you're talking about is from chapter 16, "Through the Trapdoor". Harry, Ron and Hermione have just told McGonagall about their suspicions, and asked to see Dumbledore. Since the headmaster is not there, they're unsure what to do.

'It's tonight,' said Harry, once he was sure Professor McGonnagal was out of earshot. 'Snape's going through the trapdoor tonight. He's found out everything he needs and now he's got Dumbledore out of the way. He sent the note, I bet the Ministry of Magic will get a real shock when Dumbledore turns up.'
'But what can we–'
Hermione gasped. Harry and Ron wheeled round.
Snape was standing there.
'Good afternoon', he said smoothly.
They stared at him.
'You shouldn't be inside on a day like this,' he said, with an odd twisted smile.
'We were–' Harry began, without any idea what he was going to say.
'You want to be more careful,' said Snape. 'Hanging around like this, people will think you're up to something. And Gryffindor really can't afford to lose any more points, can they?'
Harry flushed. They turned to go back outside, but Snape called them back.
'Be warned, Potter – any more night-time wanderings, and I will personally make sure you are expelled. Good day to you.'
He strode off in the direction of the staff room.

My guess would be that he saw Harry and the others standing close together, in a rather suspicious way, and potentially even heard Harry mentioning 'It's tonight' and assumed he was up to something. Snape knew very well that Harry had been caught after transporting Norbert to the Astronomy tower (which also is the reason why Gryffindor has so few points left, though these events happen a bit differently in the movie), and he believes Harry has no respect for the rules, only wanting more attention for himself, and would gladly go wandering at night again. As proven by how Snape thrives by bullying Harry in general, I have no doubt he would be very glad if he had the chance to get Harry expelled.

Though if we step out of the in-universe theories for a bit, it's quite obvious why JK wrote this scene. At this point in the book, the reader is supposed to believe Snape is the one trying to steal the Philosopher's Stone, and having him appear just when the main characters discuss this is a good way to strengthen the reader's suspicions. The reader may ask themselves: Maybe Snape wanted to get them out of the way, so he could be sure to steal the stone?

Most likely this scene if of little importance, and Snape might not have had any other intentions than to make Harry's life as miserable as possible. Potentially, he also wanted to be sure that Harry didn't intend to break any more rules. Snape is after all a teacher, and while he sure would love to punish Harry, he's also supposed to make sure the students aren't doing anything they should not. Standing inside and talking with fellow students on a sunny summer day is something most teachers would find suspicious (McGonnagal had asked them to go outside and enjoy the sunshine a few seconds earlier), and random chance might be the only reason why it was Snape.