The cable can only put a force on the mast that is perpendicular to itself: It runs over rollers which do not allow it to transfer any force in the direction of the cable.
As such, the force on the mast is exactly given by the angle between the oncoming and the outgoing cable. This force is roughly perpendicular to the slope of the cable (assuming a small deflection of the cable by the mast), especially if the cable has several supports in a roughly straight line.
The mast is setup in such a way, that the sum of this perpendicular force and its own weight is roughly in the direction of the mast itself. And since the weight of the mast is by far the smaller force of the two, you see the strong diagonal setup.
Of course, the cabins pull the cable straight down, even though the masts only take a force perpendicular to the cable. The difference is taken by the cable itself, which has a much higher tension at the upper end of the ropeway than at the lower end due to this.
As to why you need so much tension on the cable: The lower the tension, the greater the slack of the cable between the masts, and the more masts you need. It's generally just much cheaper to build two very well-anchored stations and use a high tension, than to build more masts. And that assumes that the additional masts can actually be built at sensible costs, which may not even be the case.
Also, the higher the tension, the higher the designed, directional forces of the cable relative to the forces introduced by wind, and seat-rocking passengers. And thus the safer the seat of the cable on the rollers.