Many students find eigenvectors and eigenvalues unintuitive because questions about their meaning and purpose are often left unanswered in a sea of computations. The topic only makes sense if one has a solid visual understanding of many preceding topics, such as thinking about matrices as linear transformations, determinants, linear systems of equations, and change of basis.
Eigenvectors are special vectors that remain on their own span during a linear transformation, meaning the effect of the matrix on such a vector is just to stretch or squish it. Each eigenvector has an associated eigenvalue, which is the factor by which it's stretched or squished during the transformation. Finding eigenvectors and eigenvalues can help understand the heart of what a linear transformation does, less dependent on the coordinate system.
If you can find an eigenvector for a 3D rotation, you have found the axis of rotation, making it easier to think about the rotation in terms of an axis and an angle, rather than a full 3x3 matrix. The corresponding eigenvalue would be 1, as rotations never stretch or squish anything, keeping the length of the vector the same.