4:3 was "basically" the standard aspect ratio of visual media at the time when widespread TV adoption started.

Silent films were typically shot in a 4:3 aspect ratio on 35mm film. That corresponds to a frame fitting on "4 perforations" of 35mm film with a normal lens. When films started acquiring sound there were a variety of different formats but the most successful put the sound directly on the film, in a strip that ran along the edge of the film. This took up part of the film and led to a confusing arrays of different standards before the Academy (AMPAS) established a 1.375:1 aspect ratio standard for movies with sound.

When TV was standardized in the US with NTSC they naturally chose to use a similar aspect ratio to film (since film would be used as an intermediary recording medium in many cases, and existing films were a likely source of content) but they didn't have the constraint, necessarily, of having to "make room for sound" since they had free reign to choose whatever aspect ratio they desired. The 4:3 aspect ratio ideal still had a lot of mindshare in the industry at the time and was simpler than the sort of wonky 1.375:1 academy standard.

After television formats were initially standardized the film industry moved on to widescreen formats which used a huge variety of different standards. One eventually popular standard was "anamorphic widescreen" which squeezed a widescreen image onto a full 35mm 4 perf frame which is compressed in the horizontal direction, giving an aspect ratio of around 2.4:1. Because anamorphic widescreen requires special lenses both for filming and for projection there is a cost associated with them. However, it produces high quality projections since the entirety of the film frame is used. A much cheaper though lower quality method to achieve widescreen was to just crop the frame and project it larger (using an ordinary lens). A variety of aspect ratios exist using that technique but they generally range from 1.66:1 to 2:1.

During the 1980s there were various organizations working on high definition video, and one of the influential members of the SMPTE's (Society of Motion Picture and Television Engineers) Working Group on High-Definition Electronic Production was a Dr. Powers who pushed the 16:9 format. He did a little experiment where he cut out paper rectangles of equal area of all the popular aspect ratios at the time and overlapped them, with a common center. He found that both the rectangle enclosing all of the pieces of paper as well as the rectangle representing all of their common overlap had aspect ratios very close to 16:9, and he felt that this made that aspect ratio a good compromise for supporting videos of various formats with the minimal amount of letterboxing or cropping. His arguments won out and 16:9 became the standard for HD video, driving the design of televisions and other displays since then.