Ttyjukl

Apart from the black suit and glasses, an earpiece is one part of the iconic attire of Agent Smith. Throughout the movies we can see him wearing it all the time.

But during the famous interrogation scene we see him taking the earpiece out. My question is; what is the purpose of that earpiece and what could he possibly be hearing?

The earpiece represents both an figurative and, apparently, actual connection between the Agent programs and the Matrix mainframe. You can see at multiple points in the movie, an Agent will receive information from their earpiece directing them where the humans are. They also likely receive orders the same way.

(In "real" computer programming the earpiece might be implemented as a socket or a message pipe, something like that.)

Inside the Matrix universe, it's how the Agents can appear to know things they can't possibly know, without arousing the suspicions of the plugged-in humans. When a homeless guy sees someone escape the Matrix through a phone, an agent blocks away immediately finds out because the Matrix tells him in his earpiece.

From a beneath-the-Matrix perspective, it appears that the earpiece is the only direct connection the Agents have to their controlling program. When Smith removes his earpiece, he's disconnecting himself from the Matrix just a little bit. Partly, this may be just to avoid distraction, but it's also symbolic -- he's sick of the Matrix and wants it to go away, and he's about to do something that is probably counter to his orders. So he's separating himself as much as possible from the Matrix.

Note that, when the other Agents come in to get Smith during the rescue attempt, one of them notes his earpiece is out, and remarks that Smith doesn't know that Neo and Trinity are trying to rescue Morpheus. Without the earpiece, Smith was no longer receiving intel on the humans, even though he should have been alerted immediately. This fits with the general principle that the Matrix programs are written to be as realistic as possible, even when it might seem counter-productive, because keeping the illusion intact is paramount. The Architect seems willing to even build in flaws to his program to avoid breaking the illusion.

@KutuluMike gave a great answer, but note also that the earpiece appears to exert some level of control over the agents as well. The agents seem to be incapable of completely removing the earpiece. It is one thing for Smith to pull it out of his ear, but another thing entirely for him to remove it from his person completely. As @DavidS points out, Smith removes it completely in Reloaded, with the comment that Neo had set him free. This implies that before struggling with Neo, Smith was incapable of removing the earpiece. This is, of course, when he begins down the path toward domination of the Matrix rather than submission to it. The implication is that the earpiece keeps agents in check in a more profound way than simply giving them orders. Somehow it compels them to follow the orders they are given. Even if the compulsion is built into the agent to follow all orders it receives, removing the earpiece gives them no orders to be compelled to follow.

This is more speculative, but it has always seemed to me that in addition to providing information and issuing orders, the device also works for surveillance (in the Matrix, it doesn't matter if the earpiece doesn't have a microphone on it; the appearance of the device is simply for the convenience of the people/programs in the Matrix, and making it appear to be merely an earpiece might convince them they aren't being listened to). Smith removes it in The Matrix when he begins to talk about his disdain for the Matrix, in much the same way you might hit the mute button before saying something nasty to the person on the other end of the phone. Though the Matrix itself would know everything either way, as Cipher puts it, "there's way too much information to decode the Matrix." The machines who the agents serve might not be able to isolate Smith's speech and decode it in real time without the earpiece.

There's a scene where Agent Smith has been interrogating Morpheus. Two other agents come in and one asks him what he's been doing. The other notices his earpiece is out and tells the other Agent Smith doesn't know (what's going on, such as the building being under attack). Clearly the earpiece is for communication as demonstrated by Smith being out of the loop when he takes his out and the other agent recognizing and acknowledging that.

I have to agree with the observation that the earpiece is most likely the matrix' only way of using its all knowing power and having the agents implement the corrections needed without enslaved humans getting privvy to the notion of an all powerful force changing and doing what it wants whenever it wants.

The reason why Thompson and Brown didnt know is because Smith disconnected by taking out the earpiece. The most "flexible" or inconsistent parts of the matrix by design of the matrix are the agent. They have to follow rules within the matrix but because they've got all the info of the matrix they're able control as much as is shown throughout the series.

In regards to the Hideout motel that Morpheus uses, that is what Tank or whomever is the "Operator"s Job. They are to hack into the matrix and write new code in there seamlessly to alter certain areas of the matrix that Morpheus and crew can hijack for a limited time. Which time in the matrix is not consistent with the real world so what a motel for possibly weeks is actually instantaneous when hacked. At a certain point the matrix sees the hack and autocorrects and we also see that the matrix/agents attempt to even break the rules a bit in terms of deja vu and having the brick wall cover the exit or the random people morphing into agents. Too much and they have to start the matrix over.

An all knowing all powerful matrix truly running the show with absolute power and no rules would awaken or at the very least distress the humans that are sleeping making for bad batteries. As implied when the matrix was talking to neo.