A similar and extremely famous and influential special effect is the "beyond the infinite" or "through the monolith" effect from 2001: A Space Odyssey (1969). It doesn't have any of the precise elements you ask about, but the slit scan technique used does create a particular impression of travel at supernatural speeds:
According to Wikipedia, the first time star streaks were used for a warp drive or hyperspace animation was in the movie Dark Star in 1974. The movie was directed by John Carpenter and the special effects were created by Dan O'Bannon. The entire movie is available on YouTube. Here's a link to a time stamp about ten seconds before the first use of the effect in the movie:
The effects are primitive compared to Star Wars (1977), which means the latter might be the first complete example with all of the elements you asked about.
Here's a link to every hyperspace in the Star Wars universe with a time stamp right before the first one in the 1977 film:
The first use of a similar effect in the Star Trek universe was in Star Trek: The Motion Picture (1979):
You might be interested to read the history of the concept of hyperspace at Wikipedia.
As to why it's popular, well it certainly looks cool. It doesn't hurt that Star Wars was a huge success. There have been some changes and variations. Particularly the Star Trek universe doesn't normally have star streaks, although they call their FTL travel "warp" and the in-universe physics are different from most explanations of "hyperspace". Also Babylon 5 has a different system where gates are used to allow a ship to travel in hyperspace. They show the quickly receding ship without the star streaks, and a wormhole-like effect instead.