Remember that $ds^2$ is calculated using the Minkowski metric. As @ACuriousMind is saying, you need to understand what a metric means. A metric is basically a prescription for how distances are measured in the space. So, for example, a metric for a sphere would be useful for finding the distance between two points on Earth given their lattitudes and longitudes. As long as all the coefficients in a metric are positive (like the metric for a sphere) then distances between two points are always positive and your normal intuition about the meaning of distance will work.
But the Minkowski metric has a term (the $dt^2$ term in many/most people's sign convention) with a negative coefficient. So "distances" can be positive, negative or zero. This is a signal to you that the meaning of "distance" here isn't what you are used to. Since we are talking about space-time (not space or time) "distances" do not refer to distances in space.
This is reflected in the language we use. If $ds^2 < 0$ (in the usual sign convention) then it refers to a "time-like interval". That is, there is some reference frame in which the two events separated by the interval occur at the same spatial location, and so ds can be interpreted as a time interval (in that reference frame). Conversely, if $ds^2 > 0$ then the interval is "space-like" and the two events do not occur at the same place in any reference frame. Instead, there is some reference frame where the two events are simultaneous and ds can be thought of as a distance in space.
So, the intermediate case is where the two events are neither simultaneous nor at the same location in any frame. Light leaving one event could arrive at the other event. These events lie on each others "null cones". Being intermediate between space-like ($ds^2 > 0$) and time-like ($ds^2 < 0$) they correspond to $ds^2 = 0$ even though neither $dt^2$ nor $dx^2+dy^2+dz^2$ is zero.