One way to look at this is to realize that the water is nearly incompressable, and the water in the utube behaves similar to a rope on a frictionless pulley with one side of the rope hanging lower than the other, so that the center of mass is off to one side of the pulley, causing the rope to "fall" towards the lower side. In the same manner, the column of water in the inverted utube has it's center of mass offset towards the longer side, so it also "falls" down the longer side, but the flow is opposed by friction and viscosity and the water in the two tanks, so the flow reaches a terminal velocity.
Another way is to consider the initial condition of an inverted utube filled with water and capped off. The pressure at any point within the inverted utube is lowest at the top of the utube, and increases with vertical distance from the top of the utube, so the pressure is greatest on the lowest end of the inverted utube. The capped ends of the utube rest below the surface of the water in two tanks at different heights. To simplify this, assume that the ends are equally deep in both tanks, so that the pressure just outside the ends of the capped tube is the same.
The pressure inside the caps is greater than the pressure outside the caps, so once both caps are removed, water initially tries to flow downwards from both ends of the inverted utube, reducing the pressure in the inverted utube, with virtually no expansion of the water. The pressure difference is greater on the lower tank, resulting in more of a pressure drop on the lower tank side of the inverted utube. This results in the pressure in the longer side of the inverted utube at the same height at the opening into the higher tank being lower than the pressure at the opening of the higher tank, so the pressure differential causes the water to then accelerate towards the lower tank, until wall friction in the tube, viscosity, and the water in the tanks, causes the flow to reach a terminal velocity.