Why does the Lebesgue number not exist in a open and bounded set?

Because there are too many possibilities for the open covering. For a compact set, any family of open sets that covers can be replaced by a finite subfamily of open sets that still covers. Hence the open sets can't be too small.

Without compactness, the open sets can be quite small. For example, the infinite family of intervals (1/(n+1), 1/(n-1)) covers the open bounded set (0,1). (Here n is any integer larger than 1.) No subfamily will cover (0,1), and since the sets have radius going to zero, the Lebesgue number would also have to be zero.