Prior to 2002, basically anyone could run political ads in the US relating to virtually any race or topic without making it clear who made the ad. You might see, for example, an ad disparaging (candidate 1), and it wouldn’t be clear if the ad was from (candidate 2), an advocacy group, or someone else.
This is why Congress passed the Stand By Your Ad law in 2002, which basically requires political ads to explicitly state who made the ad.
Since most people don’t pay that much attention to changes in the law, this meant that many people just started hearing “I approve this message” all the time, even though they weren’t used to hearing it before. So it is not surprising that it has spawned various memes and parodies.
For example, I could write, “I am SegNerd, and I approve of this post.” I don’t actually have any legal obligation to write that (this isn’t a political ad), but at this point, it is a “catchphrase.”