The great site Television Obscurities has the story of the television coverage, along with footage from CBS' telecast. The issue of TV coverage is a significant one; the Russians, leading the space race, were extraordinarily secretive about the whole thing (what a surprise). A major factor in the American space program was its openness - this flight, like all others, would be done not in secret, but out in the open, on television. Its success, or its failure, would be there for all to see. And there were failures, in the early days, with many of the unmanned launches. People saw rockets exploding on the launch pad, failing to reach orbit, spinning out of control - and the Russians laughed. (Khruschev called one failed attempt "Kaputnik," a pun on the successful Soviet satellite Sputnik.)
But on February 20, 1962, the world saw Glenn's success. It may have been insignificant compared to Soviet accomplishments (Gherman Titov, the second cosmonaut, orbited the earth for an entire day, far more than Glenn's three orbits), but it showed that the Americans were players in the space race. As UPI noted, "merely recording the event in a free and open way, the United States scored an enormous propaganda victory."
Here's some archival footage of Glenn's launch.