Mathematically, the intuition is simple. (This is not a formal proof, though it can easily be turned into one. It's just another dimension of intuition -- symbolic intuition.)
The price elasticity of demand at $(P_0, Q_0)$ is the infinitesimal ratio of percentage change in quantity demanded ($dQ/Q_0$) to percentage change in price ($dP/P_0$).
When the demand curve is linear, the red expression is constant: it's just the slope of the demand curve. The blue expression, however, depends on the point at which the elasticity is calculated; even for a linear demand curve, it is not constant.
In fact, if you draw a typical demand diagram, you plot the inverse demand function, with $P$ on the vertical axis and $Q$ on the horizontal axis. In this case,
- $P_0/Q_0>1$ when the point $(P_0, Q_0)$ lies above the line $P=Q$
- $P_0/Q_0=1$ when the point $(P_0, Q_0)$ lies on the line $P=Q$
- $P_0/Q_0<1$ when the point $(P_0, Q_0)$ lies below the line $P=Q$
Thus, it is easy to see why $\epsilon_d$ changes along even a linear demand curve. The inverse demand curve will cut through Quadrant I of the $QP$-plane. The closer the point $(P_0, Q_0)$ gets to the vertical axis, the larger the elasticity.