Your guess is wrong, because first of all, there is no "xy"-axis, there's only an "xy"-plane, the plane perpendicular to the z-direction.
As you note, your text describes how the matrix can be made diagonal. Not sure how well the text explains this, but the way to get the matrix to be diagonal is to go to a new coordinate system: You started out with $x$, $y$ and $z$ in "some" way. But your initial $x, y$ and $z$ might not be the "natural" axis of your object. If you go to new axes $x', y'$ and $z'$ such that the tensor is diagonal, you are now in a coordinate system where the object will rotate freely only around those new axes. 586
That's why they are called the principal axes.
The nice thing about a diagonal matrix is that matrix-vector products in them are very easy to calculate. For example, if you rotate with angular velocity $\omega$ around an axis $\vec \omega = (\omega_x, \omega_y, \omega_y)$, then the rotational energy of that is $$\frac{1}{2} \vec{\omega}^T \cdot I \cdot \vec{\omega}$$ If $I$ is diagonal, this simply becomes $$\frac{1}{2} \omega_x^2 I_{xx} + \frac{1}{2} \omega_y^2 I_{yy} + \frac{1}{2} \omega_z^2 I_{zz}$$
It is, however, correct that $I_{yy}$ is the moment of inertia for rotation around the $y$ axis. The off-diagonal elements would come into play if you don't go to the coordinate system that makes $I$ diagonal and then look at rotations around axes different from the coordinate axes. For example $\vec \omega = (\omega, \omega, 0)$. There, the rotational energy would be $$\frac{\omega^2}{2}(I_{xx} + I_{yy} + 2 I_{xy})$$.
So if you want to compute the rotational energy in a coordinate system where $I$ is not diagonal, you get all those pesky off-diagonal matrix elements in there, cluttering up your expression, whereas you can get rid of all of them if you transform the coordinate system so that they are $0$.
This diagonalization, btw, is something that comes up very often in all fields that deal with matrices, just because it makes dealing with those matrices so much easier, and the remaining diagonal entries (the "eigenvalues" of that matrix) contain a lot of useful information about the nature of the matrix. 499