Get a sheet of rubber. Grab it in both hands and stretch it. Inspect your sheet and find a line on the sheet that you could draw in with a marker and when you stretched the sheet the line would grow and shrink, but would not change what it was pointing at (probably a line from one of your hands to the other, in this simple example) That is an eigenvector of your sheet stretching transformation. The eigenvalue is how hard you're stretching the sheet.