TL;DR : No official explanation, but we can speculate Leeloo represents humanity and its potential.
Ok, some of this is supported by comments by Luc Besson, but it's mostly based on speculation & discussion among fans. Besson purposely left the film's various meanings ambiguous because he wanted viewers to make their own conclusions. We're therefore left to infer from dialogue in the script, film & novelization.
The Fifth Element is humanity, and Leeloo is the perfect specimen representing that element. Basically, Leeloo is the "perfect" human, with her DNA containing the blueprint for everything that humanity has the potential to become. It's never stated in the film, but there has been speculation among fans that Leeloo is, in fact, the first human being - the genetic "template" from which all other humans & animals on Earth were created. Simpler versions, yes, but based on the same DNA. This speculation comes not only from her tightly-packed DNA, but also from her weapon against evil being the "Divine Light of Creation".
Here are some quotes from the film that support the "perfect human" theory:
In 1914, the Professor (when reading the inscription on Leeloo's chamber):
The Professor keeps reading over the inscription.
PROFESSOR: "..this perfect person.. this perfect being.. I do not understand this.. perfect?"
Then later, when Leeloo is resurrected:
MUNRO: Have you identified it?
MACTILBURGH: It's not that easy.. we've never encountered anything like it before.. you see, normal human beings have 40 DNA memo groups, which is more than enough for any species to perpetuate itself. This one has 200,000.
MUNRO: Talk English, Doc.
MACTILBURGH: This cell is like a huge library. It has infinite genetic knowledge stored inside. Almost like it was...engineered.
Leeloo is not "perfect" because she kicks ass and looks good doing it. She is perfect from a biological standpoint.
The theory of humanity in general being the Fifth Element is largely supported by the fact that the great evil (the black planet) existed to destroy all life, yet its first target was Earth. There are obviously other planets in the universe with civilizations on them, so why make a beeline straight to Earth? Because humanity, or what it can potentially become at some point, is the greatest threat. It also backs up the concept that Leeloo had to embrace her humanity before she could destroy (attack?) the great evil. If anything, the overall message of the film (and Leeloo's self-discovery) is "Humanity has its flaws, but also the potential to be more."
As for the statue at the beginning, the "statue" taken by the Mondos was actually Leeloo's stasis chamber - like a "coffin" of sorts with her inside. Later, when returning to Earth, we can assume that the Mondos had brought her out of stasis and she was wearing one of their environment suits when the ship was shot down. This explains why her hand was in a Mondo glove, and she would have been the natural choice to carry the case (even though the stones weren't actually in it at the time). Either its mere coincidence that her hand was the recovered material (and not a Mondoshawan), or it was because of her enhanced durability that even the hand was recovered.
UPDATE: According to AMC Story Notes (the trivia that can appear on-screen before a movie), Director Luc Besson said the real fifth element is "life, love, and art."