A diagonal matrix has numbers only along the diagonal and zeros everywhere else. Apply one to a vector (x, y) in the plane and the answer is (λ₁x, λ₂y). The x is multiplied only by λ₁ and the y only by λ₂, and the two never mix. That is why the grid in the animation above only stretches along the axes and never tilts or turns.
Because nothing mixes, the shape survives repetition. Applying D n times gives Dⁿ, which is just λ₁ⁿ and λ₂ⁿ sitting on the diagonal. You never have to walk through the intermediate steps: the position after n applications is (λ₁ⁿx, λ₂ⁿy), written in one go. Raising an ordinary matrix to a power is hard work, but for a diagonal matrix multiplication turns into adding exponents.
The long run therefore depends on the factors alone. A direction whose factor is greater than 1 in absolute value grows without bound, one below 1 is drawn into zero, and at exactly 1 nothing moves. A negative factor flips the direction at every step, and a factor of 0 flattens the plane onto a line in a single step, with no way back. There the determinant is 0, so no inverse exists. All five cases are waiting in the examples.