What was the cultural impact of the invention of the radio?
The previous answer does a great job of talking about the cultural impacts of radio that are more economic and political. I'd like to add one other impact.
The beginning of commercial radio is associated with the 1920s, which was a time of great cultural upheaval in the US. Along with other media, such as movies, and other technology, such as cars, the radio helped bring in a new era in which Americans had many more entertainment options. This led, historians say, to a new culture whose nature is captured in the terms "Roaring '20s" and "Jazz Age."
This new culture was typified by the flappers and their new, more hedonistic values. The new culture inspired backlashes as well, such as the rise of religious fundamentalism and of the KKK (which at that point was more focused on nativism and anti-newness than it was anti-black).
The radio didn't do this stuff all by itself, but it was an important factor.
The most signficant thing is that it suddenly united people in a collective identity of "here and now." (Consider the immediate popularity of Rooservelt's "Fireside Chats.") People could get information 'on the spot,' hear all the unstated information in interviews and debates (hesitation, pitch of voice, cadence of sentences) so important in deciphering the real import of what was being said, and have a better global appreciation of what was actually happening in current events. (The other side of the coin is that public opinion could be manipulated better than ever!) Also, a tremendous speculative market was open for advertising, which was of course increasingly exploited over the years. Radio was in such a way the first real means of 'tele-' (meaning 'long distance') communication.