Multiply almost any vector by a matrix and it points somewhere new and comes out a different length. A tiny few vectors don't — and that exception is the whole reason eigenvalues show up in vibration, control, stress analysis, and data science.
A matrix is more than a grid of numbers — it's a linear transformation. Feed it a vector and it hands back a new vector: rotated to a new direction, and stretched or shrunk to a new length. That's what a matrix does. Almost every vector you could pick gets bent off its original line. But every square matrix has a small, special set of directions where that doesn't happen. Send a vector in along one of those directions and it comes back out pointing exactly the same way (or exactly the opposite way) — only longer or shorter. Those special directions are the matrix's eigenvectors. The amount each one gets stretched, shrunk, or flipped is its eigenvalue.