For three days Jesus was "with the dead".. meaning that he was not with God.
This underlines that he was separable from God, and under his own authority, and it contrasts to Jesus' being the only way to eternal life (with God).
It is common for people to talk about "Hell" as his destination for these days, and most churches even recite the Apostles Creed to this effect. But if you consider the text of the creed and the Hellenistic world at the time it isn't the proper conclusion.
descendit ad inferos, tertia die resurrexit a mortuis,
The latin word "inferos" depicts the Hellenistic premise of a "land of the dead" ... but the inference of this into the Pan-European "Fiery" Hell is so pervasive that you'll find that if you enter that latin line in its entirety into a translator you'll come up with "hell"... but if you enter just the word "inferos" you'll come up with something like "underworld".
Hellenistic culture (which we find as a significant audience to the ministry of Jesus & the apostoles) had a cultural understanding of "Hades" as an "underworld" where everybody went to journey until they reached Elysium or Tartarus.
The first stage of Hades, Asphodel, was effectively a place of wandering. This parallels the Judeo-Christian premise of where humans went after they left the Garden (Nod, wandering) and what the Israelites had to wander between slavery (as the Gospel notes we are slaves to sin) and the fulfilled promise.
But Jesus speaks to the thief next to him about "paradise" which parallels the concept of Elysium. The audience at the time would understand Jesus to be telling the man he *will not go to a place of Torment, but to a place of joy.
Why three days? Many answers in this thread have addressed this well: rotting bodies, plague of Egypt, "because he said he would", the rebuilding of the temple.... and these are all great examples of the underlying premise of the symbolic nature of "three" in Hebrew numerology to mean something like "a good amount, but not too much"
This is contrast with the importance of things like "seven" to mean "a whole lot!" and Jesus' use of "seventy-seven" to mean "way more than a whole lot!"