In intuitive terms:
Let's say I ask you to throw a dart at a dartboard. To keep things simple, assume the face of the dartboard has an area of 1 square meter, and that you are equally likely to hit any spot on the dartboard. What is the probability that the dart will land at a particular point on a dartboard?
We can approximate this by discretizing the dartboard and thinking of it as a bunch of tiny squares that are 1 millimeter on each side. In that case, the probability of landing in a specific square is 1 in a million. But this is an approximation, and we're not really happy with it, so perhaps we can divide the dartboard into "points" that are 1 micrometer on each side. Now that we have a finer granularity, the chances of hitting a specific point are 1 in a trillion.
Of course, this is still an approximation, and we want to be exact, so let's look at individual molecules on the dartboard's surface. What is the probability of hitting a specific molecule? It'll be a tiny number -- maybe something like 10^-30. In the pursuit of accuracy, we can go beyond molecules to atoms or even to quarks. As we get closer and closer to defining a point as being infinitesimally small, the probability of hitting that point gets infinitesimally close to 0.
In a continuous distribution, a "point" is the same as a "point" the dartboard -- it's dimensionless. You can get non-zero probabilities if you approximate the point with some interval, but as the interval shrinks your probability will approach 0.
In math terms:
Probability of an event = (# of outcomes that lead to that event) / (total # of outcomes)
In a continuous distribution, there's an infinite number of outcomes, so the probability of a specific outcome is 1/infinity, which is 0.