Light has nothing to do with this.

The point is that mass bends space-time. The amount of bending is dependent on the size of the mass and on the distance from the mass. Even though the Sun is incomparably heavier than the Earth, it is also MUCH farther away from you. So, space-time around the Earth is curved much more towards the center of the Earth than it is towards the center of the Sun. In the mattress analogy, consider a large mattress, with a bowling ball and a car sitting on it. The car will obviously bend the mattress much more, but if you're close to the bowling ball, you'll still fall towards the bowling ball first before both you and it fall towards the car.

So, say you're in an airplane moving directly forward, with the Sun just overhead (and the Earth obviously just below you). The Earth curves spacetime towards it a lot in this area, while the Sun curves it towards itself just a little bit. The overall curvature is such that time still moves more for the bottom of the plane (closer to the Earth) than the top of the plane (closer to the Sun). So, the bottom side moves a little slower than the top side, but the structural integrity of the plane pulls the top side towards the bottom, causing a slight motion towards the Earth - gravity [note that the GP's explanation got the signs a little wrong - time flows slower, not faster, closer to a big mass]. Conversely, if the Earth disappears from the picture and only the Sun remains, now the top part of the plane will move slightly slower, pulling the bottom part towards it, and thus towards the Sun.