As long as cameras have been around, courts have had the discretion to ban them from the courtroom. The origins of the modern rules prohibiting camera use came from Federal Rule 53, adopted in 1947, which provides:

Except as otherwise provided by a statute or these rules, the court must not permit the taking of photographs in the courtroom during judicial proceedings or the broadcasting of judicial proceedings from the courtroom.

In 1972, the rule was changed to add specific language banning television coverage. The rule was created within the context of criminal trials, but the Supreme Court, in their discretion, chose to apply it to Supreme Court proceedings as well. That makes it a bit tricky to find the exact reasoning behind their decision because most of the analysis I know of from the time only addresses criminal trials. According to Moore's Federal Practice, the ban on cameras was based on primarily on a fear of uncontrolled publicity. There was great concern in criminal trials that the presence of cameras could have an adverse effect on how the trial proceeds and could potentially bias the jury or subject the defendant to humiliation. See, Chandler v. Florida, 449 U.S. 560 (1981) (discussing the possible adverse effects of cameras in a criminal trial).

The origin of this decision dates to 1935 in the case of State v. Hauptmann, otherwise known as the Lindbergh Baby trial. The trial received such wide media coverage that the court became concerned that it was a distraction. The defendant appealed his conviction on the grounds that the presence of so many cameras and media members in the courtroom biased the trial. He lost, but in 1937, the American Bar Association amended their Canons of Judicial Ethics to recommend barring photographs and videos of trials (ABA Canon 35).

As for the Supreme Court itself, I don't have a lot on why they specifically chose to follow this rule, but in recent years, Justices have made many comments both in favor of and opposed to it. The primary argument against camera use has been to preserve the sanctity of the proceedings and prevent Justices from showboating to the cameras or letting public perception affect their judgment. As of right now, the court is split on the issue. Chief Justice Roberts and Justices Alito and Kennedy are known to be against changing the rules, while Justices Ginsburg, Sotomayor, and Kagan support allowing cameras. Justices Breyer, Gorsuch, and Kavanaugh have not voiced an opinion I'm aware of. So it's unlikely to change as long as Roberts is Chief Justice.

I'm sorry I don't really have sources apart from the laws and cases themselves. I wrote about this issue in law school and am mostly going off of memory. If I find my sources, I'll be sure to come back and put them in.