"the xz plane is perpendicular to the y-axis, and the yz plane is perpendicular to the x-axis"
We are given a three axis model of space; the three axis are each perpendicular to the other (imagine looking at a corner of a cube, you see three lines intersecting at the corner - those three lines are perpendicular to each other).
The xz plane is the infinite flat space (ie plane) that is on both x and z axis. Since each of those axis is perpendicular to the y-axis, the xz plane is perpendicular to the y-axis as well. Similarly the yz plan is perpendicular to the x-axis.
To see this visually take your cube and pick a corner, on each of the three lines (ie edges) that forms the corner assign one of the labels x, y, or z. Now you can see that the plane defined by the lines x and z (the xz plane) is in fact perpendicular to the line labeled y.
Your understand is right but you leave out enough to leave it unclear to the reader if you actually understand it :) Since you are asking one must assume you might not. Hopefully this answer crystalizes your understanding one way or the other. IMO this question fits better in math.SE.