He's being investigated, by a Senate committee, for the collateral damage he has been causing. The specific hearing we see in the movie was prompted by the incident in the desert near the start of the movie.
First off, it's important to note that it's not a court of law. It's a Congressional investigative hearing called by the Senator to determine if there's a problem that the US government needs to solve. The Senate committee (in this case) does not have the authority to arrest him (though any evidence they uncover could be used as part of judicial proceedings later). But mostly, the purpose of those hearings is to determine of there's some legal remedy needed to prevent the problem from happening in the future.
Second the hearing was investigating the events surrounding Superman's actions. Senator Finch was already investigating Superman's actions as far back as Man of Steel. For example, the man in the wheelchair was injured in the destruction of the Wayne Building was attempting to testify that it was Superman's fault that he was paralyzed.
The issue seemed to come to a head with the events surrounding Lois' rescue from the terrorist in the desert. Although it's not very clearly explained, if you listen to the witnesses closely enough, you will see one Arabic woman giving testimony that seems to implicate Superman in all of the murders that happened during that scene. (She describes a "loud boom" and then later seeing "bodies everywhere"). All of the men who were murdered by the private security force are being blamed on Superman, because no one else was alive to contradict those claims.
Ultimately, we are supposed to believe that this was all
a plot by Lex Luthor to discredit Superman by implicating him in a murder.