You have to think of where pressure is lowest and where it is highest.
The wing is forcing air downward along its entire span, creating a low pressure area above the wing and high pressure below it. Higher pressure air wants to move in to fill the low pressure voids. The ambient air above and away from the wing moves downward toward the low pressure zone. This accelerating air causes lift, and the downwash.
At the wingtips, in addition to ambient air above and below, the accelerating air encounters ambient air to the outside, so the ambient air moves inward toward the low pressure zone. The high pressure below the wing pushes outward toward the ambient air. This causes a rotation. The rotation is the beginning of the vortex. It begins at the wingtips because that's the first place it can occur.
As the wings move away they leave a sheet of low pressure behind them. The low pressure zone is pulling ambient air downward and inward from above whereas the high pressure is pushing downward and outward into the ambient air. As each parcel of air moves toward the voids, the parcel behind it moves in to fill in where it was, so the whole process expands outward as it involves a larger area. The rotation that began at the wingtips spreads outward creating the rotating funnel shape.
You can see this in the photo below. As the vortices spread and the moisture of the engine contrails get caught in them you can see how they move. The inner contrails are being pushed down and out. At the top the contrail the parcels of air are being pulled in and down toward the void and the outside is being pulled up and around to fill in where the first parcels left. The sheet in the center is the low pressure area. The low pressure is causing water to condense into a fine mist that can be seen by the sheen near the top of the photo.
Zooming in on the wingtip you can faintly see the vortex forming at the tips.