Why transform normals with the transpose of the inverse of the modelview matrix?

I am working on some shaders, and I need to transform normals.

I read in few tutorials the way you transform normals is you multiply them with the transpose of the inverse of the modelview matrix. But I can't find explanation of why is that so, and what is the logic behind that?

Take a look at this tutorial:

https://paroj.github.io/gltut/Illumination/Tut09%20Normal%20Transformation.html

You can imagine that when the surface of a sphere stretches (so the sphere is scaled along one axis or something similar) the normals of that surface will all 'bend' towards each other. It turns out you need to invert the scale applied to the normals to achieve this. This is the same as transforming with the Inverse Transpose Matrix. The link above shows how to derive the inverse transpose matrix from this.

Also note that when the scale is uniform, you can simply pass the original matrix as normal matrix. Imagine the same sphere being scaled uniformly along all axes, the surface will not stretch or bend, nor will the normals.