First of all, Solon and Socrates didn't leave any writings, and Herodotus never once mentioned Atlantis. Early references to Atlantis come solely from some of Plato's dialogues. The character Socrates who exists in Plato's writings is a literary invention who is partially based on his real teacher.

Since the story originates from Plato and has no close parallels with anything else we know if, it seems reasonable to assume he invented the story as an allegory to prop up one of his arguments. He did this a lot. Ever heard of the Allegory of the Cave? Or The Ring of Gyges?

Some historians believe the story is not one hundred percent invented, and may have been inspired by something that actually happened. This doesn't mean there really was a massive continent-sized island with an advanced civilization. This means it could have been inspired by the true story of the Minoan outpost on Santorini/Thera which was destroyed by a volcanic eruption. Or it could have been inspired by the destruction of the city of Helike in central Achaea (in the Peloponnese) in 373 BCE, shortly before Plato wrote about Atlantis. In fact, I recently volunteered at the Helike Project to excavate the city and the archaeologist in charge of the expedition does believe this city is the inspiration for Plato's fictional Atlantis.