I don't know if this is correct, but this is how I made sense of it, I think this a very intuitive explanation.
Imagine a car moving (from left to right) in a straight line with constant speed of 100MPH, and at x=0 the breaks of the car are engaged. Naturally the car will start to decelerate until it comes to rest (speed = 0 MPH).
The car will certainly come to rest, but the question is:
At what distance D [meters] (with respect to x=0) will the car come to rest? (This value D is not known apriori and can vary from one experiment to the next by a myriad of factors, so we can ask probability questions about the variable D).
Since distance is modeled by real numbers, the variable D is continuous. If we ask:
what is P(D=6.354), the probability that it comes to rest at specifically D = 6.354 meters?
then, since the probability distribution of D is continuous, the answer is P(D=6.354)=0. This probability is zero.
Notice that there is nothing special about the number "6.354". That is, for any real number r the probability that D is exactly r is zero, i.e. P(D=r) = 0. This is simply a consequence of how continuous probability distributions are defined.
Regardless, we know that the car will certainly come to rest and that this happens at some specific distance d (where d is a real number). So even though P(D=d) = 0, it does not mean that coming to rest at D=d was impossible; clearly it was not because the car does come to rest at d.