classical mechanics - Physical meaning of the moment of inertia about an axis

In the context of rigid bodies, the inertia tensor is defined as the linear map that takes angular velocity to angular momentum, that is, the linear map $I : \mathbb{R}^3\to \mathbb{R}^3$ such that

$$\mathbf{L}=I\boldsymbol{\omega}.$$

Now, given one unit vector $\hat{\mathbf{n}}$ characterizing the direction of a line, one can define

$$I_{\mathbf{n}}=\hat{\mathbf{n}}\cdot I(\hat{\mathbf{n}}),$$

which is the moment of inertia about that axis.

In that setting, if $\boldsymbol{\omega}= \omega \ \hat{\mathbf{n}}$ one gets, for instance, the nice looking formula for kinetic energy:

$$T = \dfrac{1}{2}I\omega^2,$$

where $I$ is the moment of inertia about the axis of rotation.

Now, although I grasp mathematically what is going on, I have no idea whatsoever about the physical meaning of the moment of inertia about an axis.

What is the physical meaning of the moment of inertia about an axis? What it really is, and how this physical significance relates to the actual mathematical definition I gave?