We know that for a continuous random variable, the probability at a particular real number is 0. But the probability over a range may be a non-zero value that lies between 0 and 1.
I know that mathematically, since we define probability using integration, the value for a range may become non-zero.
But how to understand intuitively that the sum of zeros become a non-zero value?
I think you're going at this the wrong way. Here is my intuitive take, implicitly assuming some vague niceness like a continuous pdf.
Clearly, the probability for the variable to be in a given (positive-width) interval is (or at least can be) non-zero. That's the fundamental principle at work here. That's where we start.
From there to the probability of getting a particular value, there are no conceptual issues. Clearly the probability of getting the exact value $x$ must be lower than landing in an interval around $x$. Those intervals can be as narrow as you want, and for narrow enough intervals, the probability is roughly proportional to the width of the interval. So clearly the probability of $x$ can't be any positive number.