To get intuition, I recommend starting not with the definition, but with the problem you're trying to solve. Namely, given a rigid body $B$, with a certain angular velocity $\vec{\omega}$ about its center of mass, what is its angular momentum $\vec{L}$? Define the "moment of inertia" to be the property of $B$ that determines the value of $\vec{L}$ given $\vec{\omega}$.
Defined this way, the "moment of inertia" could in principle be some horribly complicated function that depends in some nonlinear way on the shape of the rigid body. After all, there are infinitely many possible input vectors $\vec{\omega}$, and a priori, they could give rise to values of $\vec{L}$ according to some crazy calculation that requires you to keep explicit track of the exact position of every particle in the rigid body.
What saves us from this nightmare is that the output vector $\vec{L}$ depends linearly on the input vector $\vec{\omega}$. How can we see that this true? Well, it's pretty intuitive that if you double $\vec{\omega}$ then you'll double $\vec{L}$, so that's a good start. What's less immediately obvious is that if $\vec{L}_1$ is the angular momentum corresponding to a certain angular velocity $\vec{\omega}_1$, and $\vec{L}_2$ is the angular momentum corresponding to a certain angular velocity $\vec{\omega}_2$, then the angular momentum corresponding to $\vec{\omega}_1 + \vec{\omega}_2$ is $\vec{L}_1 + \vec{L}_2$. But intuitively, you can convince yourself of this if you imagine the special case where $\vec{\omega}_1$ points along the $x$-axis and $\vec{\omega}_2$ points along the $y$-axis, and you calculate the angular momentum of an infinitesimal element of the rigid body at $(x,y)$ rotating first according to $\vec{\omega}_1$, then according to $\vec{\omega}_2$, and finally according to $\vec{\omega}_1 + \vec{\omega}_2$. And if linearity holds at the infinitesimal level, then it must hold of the whole rigid body, because integration is linear.
Once you've established that $\vec{L}$ depends linearly on $\vec{\omega}$, you're basically done; that's what a rank 2 tensor is—something that gives you an output vector from an input vector in a linear way. So the moment of inertia isn't an incomprehensibly complicated function of the rigid body after all; because it's a rank 2 tensor in 3 dimensions, it's determined by (at most) 9 numbers. 499
These numbers capture the body's resistance to torque in three independent directions, and linearity lets us derive its resistance to torque in any direction.
By the way, this way of thinking also helps one see that the moment of inertia is some physical property of the object that enjoys an existence independent of our choice of coordinate system. The angular velocity and the angular momentum are physical, and the relationship between them depends just on the physical structure of the rigid body. Thinking of the moment of inertia tensor as a coordinate-independent linear relationship between coordinate-independent quantities (vectors) lets us derive the transformation law when we change coordinates (as opposed to taking the transformation law as the definition of a tensor, which I have always found to be very confusing.)
One more remark. The aforementioned exercise involving an infinitesimal element of the rigid body helps us see why the moment of inertia tensor isn't described by just a single number. That is, $\vec{L}$ isn't just a scalar multiple of $\vec{\omega}$, because the same angular speed about two different axes will give rise to different magnitudes of the angular momentum, depending on how far the mass is from the axis of rotation. So we need more than one number. Now, as you are no doubt aware, the moment of inertia tensor is actually a symmetric tensor, so you don't actually need 9 numbers to describe it; 6 numbers suffice. This symmetry doesn't follow just from the general fact that $\vec{L}$ depends on $\vec{\omega}$ in a linear way; it is a special fact about the mechanics of rigid bodies. But I think the main obstacle to intuition is grasping what a rank 2 tensor means physically, and that is what I have focused on here. 499