The observed shift was due to the combined effects of the motion of the Earth and the finite speed of light.
Imagine that you are standing outside during a rain shower where the drops of rain are falling exactly vertically. This is confirmed by holding a long pipe, straight up, and seeing that the raindrops don't hit the sides. If you start moving the you need to tilt the tube in the direction of motion to get the drops to hit the bottom of the tube. The required angle depends on the speed of the raindrops and the speed of your walking. Bradley realized that this accounted for the strange timing of the observations because the largest angle occurs when the Earth's motion is perpendicular to the line to the star. The observed shift also implied a speed of light that was in good agreement with the known value of the speed of light.
This story has yet another wrinkle. If the raindrop analogy is taken literally then the speed that the drop falls in the tube should effect the angular displacment. Since the speed of light is different in a medium than in a vacuum a telescope was built that was filled with water by George Airy. When Airy repeated Bradley's experiments with this telescope the expected change was NOT seen. The reason for this remained a mystery until Einstein's Theory of Special Relativty showed that velocities add strangely when near the speed of light. Strangely, there are still people that claim the null result of Airy is proof that the Earth is stationary.