Let me put it another way (to compliment my other response to this post).

If you throw a ball and you aim at the target directly, gravity will pull it down and you won't reach your target. Instead you throw it at an angle up so that when it reaches the target distance it will also be at the target height.

If you start with the same initial velocity (the strength of your arm) and you want to throw it further you need to adjust your angle up higher.

If you are on a train however, and you throw it with the same velocity it will travel much farther because the velocity of the train will add to the strength of your arm. You then have to adjust the angle lower to hit your target.

This isn't just an analogy to what the probe is doing, this is exactly what the probe is doing for exactly the same reason. They are "throwing" the probe, with the strength of a rocket, on a moving train (the Earth) to hit a moving target (Mars). Given the total strength of the arm (rocket) plus the velocity of the train (Earth) they calculate the angle they have to launch at to hit the moving target. This gives the start of the curved path, which will be a much longer path (along the path) than that calculated if you were to draw a ruler line from the thrower to the target.

The total velocity will be the strength of the arm (rocket) plus the velocity of the train (Earth) relative to someone standing on the ground (the Sun). But from the train (Earth) it will look much slower, because the train is moving in the same direction.