It's pretty simple, really. A league is an old-fashioned measurement of distance that's roughly equivalent to three miles. 3 x 20,000 = 60,000 miles. This is the distance Aronnax, Nemo, and company travel under the sea, not the depth they go while traveling.
To put things in perspective, 20,000 leagues is about a quarter of the way to the moon when the moon's closest to earth (Verne wrote about that trip in his book From the Earth to the Moon). On the other hand, the deepest part of the ocean, found in the Pacific's Mariana Trench, is less than seven miles. Yeah, it's a pretty darn long way.
So, we think the title serves as a reminder of how epic and extensive the gang's journey is. And the whole "under the sea" part of the title is important because it points to what separates the Nautilus from a regular old boat; it can travel beneath the waves and give its passengers a fish's-eye-view of the ocean.
If you want to avoid confusion, perhaps it's best to look at the original French title, 20,000 Lieues Sous les Mers—20,000 Leagues Under the Seas. That plural, "seas," makes it clear that Verne's writing about a trip through the world's seas, not into their darkest depths.