"I am sixteen years old."
From what I've learned from other comments on similar questions, the difference would be like "I am sixteen" (Jyuu roku desu) vs "I am sixteen years old" (Jyuu roku sai desu). The latter sounds more formal to me, can't remember if the others said it was or not.
There's likely more specifics on the differences/why the 'sai' is used but this is what I managed to gleam from the others.
That's a pretty good guess, but I'm afraid your understanding is a bit incomplete.
The さい (歳/才) must be used here for the sentence to mean "I am sixteen years old."
In English, because of plurality agreement, the "years old" part is very strongly implied when we say "I am sixteen", but the same thing can't be said of the Japanese sentence. In Japanese, さい is basically a counter for years of age; without it, what you are "counting" when you say "sixteen" is too ambiguous. 「十六です」 would be more like saying you (or whatever the subject is) are the idea of the number 16 itself, rather than sixteen of something.