In The Avengers, I've watched and rewatched this scene many times but I still can't find a good reason as to why Banner would reach out and grab Loki's scepter. Was it some sort of plot point that we'll get later?
Bruce Banner is man with, as Tony Stark says half in jest, "breath taking anger management issues".
Although not really demonstrated all too much in the Marvel Cinematic Universe, Bruce Banner is a man with multiple reasons to be angry at his own existence - he has a mostly uncontrollable monster within him at all times, he's constantly terrified of injuring people and he's been forced into hiding by various US Government agencies, distancing him from the people he loves. In the comics, it's been demonstrated that he has some deep psychological problems and self esteem issues (see The Ultimates), and that he had trouble with his anger before he first turned into The Hulk.
So when Bruce Banner realises that;
- Loki is trying to use him (specifically, a part of himself that he resents) in part of a plan to take over the Earth
- the people he has been bonding with for the last day or so are terrified of what he could become
- Fury is using the Tesseract to create weapons of mass destruction
all in the space of a few minutes, he begins to show some of the anger that he tries to keep under control.
We find out in Avengers: Age of Ultron that Loki's sceptre contains one of the Infinity Stones, the Mind Gem, which is capable of manipulating people's emotions and fears, at least to some degree. It's proximity to the group of arguing people is no coincidenece - during the scene in question, the sceptre is slightly glowing as the camera focuses in on it, indicating that the sceptre is beginning to make people more rash then they otherwise would be, demonstrated perfectly by the increasingly heated arguement between Steve Rogers and Tony Stark.
As such, Bruce Banner gets more worked up than usual, to the point that he ends up picking up what he knows to be a very dangerous weapon without even seeming to realise it. When he later turns into the Hulk and roars, we see Loki react to the noise with a sly, self satisfied grin, indicating that this was all part of his master plan.
It all comes down to the idea that Bruce Banner has very little control over his emotions, so when his emotions are amplified by a magical sceptre, he acts irrationally and without thinking. It's there to demonstrate that Bruce Banner is quickly losing control of his emotions, raising the tension for the audience and the characters in the room - because we all know what happens when Bruce Banner gets angry, and you probably won't like it.
It wasn't exactly on purpose. Everyone was getting angry and it of happened in the heat of the moment. From the filmmaker's perspective, it's a way of reminding the viewer that the reason everyone is so mad is because Loki is somehow influencing them, trying to tear the "team" apart. There isn't really much of an in-universe reason.
This was mentioned in the director's commentary.
Arguably it's a movie maker's point rather than a plot point. They need to distract from the fact that the trace has just found the location and that if they were paying attention rather than bickering, they could already be making strides towards resolving it but as they are more concerned about Banner with the scepter it's missed. This sets up realization this was Loki's plan and a moment of calm before the bomb goes off. It's a false dawn cos of course Hulk then manifests himself moments later as intended but the job was done, their eyes were off the ball at the crucial moments and on Banner.
His reaction by putting it down shows that he knew immediately he'd been played and he was back "in the room" and in control. The others though had seen the glimmer, so even had the attack not happened the damage was arguably done. To me it was that he reacted again to being "asked nicely" rather than a conscious decision. Cap asks him nicely to put the Scepter down and he does, just as Widow "asks nicely" for him to come in and even the little girl asking nicely for him to come to the sick person. It seems Banner is so used to no contact that basic courtesy shown to him makes him react favorably to them, even Stark's joking "suiting up" stuff is nice rather than goading.
One thing is sure, it's not the Scepter itself, but it perhaps amplifies Loki's powers. Banner is immediately showing signs of Loki "in his head"/migraine when he even sees Loki sans staff... the Scepter probably just allows Loki to be "in that room" even when he is decks away.
Obviously the scepter has the capacity of manipulating the MIND...because it has the MIND gem...that is why Loki was able to control Hawkeye through the force of the mind gem........and is no wonder that Bruce Banner started to get a little angry in the argument and grabbed the scepter as a reflex influenced by the power in the stone.