The way I have explained the fundamental theorem of arithmetic in the past is by establishing what it means to be prime (has exactly two positive divisors) and then having students construct factor trees (where the prime factors at the end are circled).
The prime definition avoids some of the caveat-language otherwise seen ("a number only divisible by itself and 1, but not 1") and the factor trees are fun to do with a number like 72, after which you can ask different students to draw their trees on the board to compare and contrast.
One important observation to emerge is that when you create a factor tree for a natural number, although different numbers may appear along the way (and although some may not appear at all -- for example, the commonly used approach never reveals the factor of 1) the circled numbers at the bottom of the tree are always the same. And, in fact, this is the fundamental theorem of arithmetic:
Fix a whole number: All of its factor trees have the same numbers at the bottom.
The next approach I take is to give students a more formal definition of the fundamental theorem of arithmetic, and then ask them to think about different metaphors that capture this idea. The most common ones I have seen are DNA and fingerprints, but having them create analogies for a mathematical idea is a powerful tool for instruction and learning. Once students come to see prime numbers as something like building blocks that uniquely define each whole number, I think that the question of why it works like this can be quite successfully postponed till a much later course.
All of this is to say, I believe the why is best answered by developing one's intuition around the theorem, and that a more careful answer than that might be best staved off till a course that involves formal proof writing (at which point the existence of prime factorization is shown easily to students who can think inductively -- factor the number, if any number in its factorization is non-prime, then that means it can be written as the product of strictly smaller factors, so continue till you have all primes -- and then use something like Euclid's lemma to show uniqueness). Incidentally, the existence part of the theorem is also a very good example of when you want strong induction in your toolbox.
Lastly: You mention that some students "know what the theorem means," and so I thought I'd direct you to a nice paper that discusses this in the context of younger students (ages 11-13):
Griffiths, M. (2013). Intuiting the fundamental theorem of arithmetic. Educational Studies in Mathematics, 82(1), 75-96. Springer Link.
(You may be interested in some of its references, as well.)