real numbers are a closure of the set of rational numbers under the limit operation. It just happens that the closure is uncountable.
It is pretty standard device in mathematics - having a set and an operation, to produce and explore another set built as a closure of the first set under the operation. Seems pretty intuitive to me :)
What is really unintuitive, puzzling is the starting point of it all - natural numbers... 1, 2, 3... why there is a one stone, a one star, ... why this discretization?