In a liquid like water, the pressure acts in an isotropic way.
That being said, imagine a slice of water in the middle tube; what are the forces acting on this slide ?
The force exerted by the pressure on the left side, and the one on the right side.
The one on the left depends and the height of the water column in the left pipe. The one on the right depends on the height in the right pipe.
If you want equilibrium, both have to be equal. Therefore the heights have to be equal.
About the pressure: pressure has dimension of force divided by surface, in common units: $N/m^2$.
The column of water on the left pipe exert a force, due to its weight (gravity) that is $g \rho S h$, where $\rho$ is the volumic mass, S is the cross section of the pipe and h is the height.
But the pressure is $g \rho h$, thus independent of the cross section of the pipe. This force (gravitational) acts downward, but the fluid make it acts in an isotropic way, thus being directed from left to right on the slice of water (see above).
A good schematic explanation is available in hyperphysics.
Altough the diameter of the left pipe is bigger, the force exerted on the "slice" of water is not higher because the pressure on a given infinitesimal volume depends only on the height of the column of water above it. Imagine two simple straight vertical pipes filled with water and with equal height, one with a large diameter and the other with a smaller. It is true that the force on the bottom of the big one is higher, but the pressure will be the same, because the force acts on a larger surface.