In what little knowledge of space and rockets I have, it seems to me that the further the object your aiming for, the more fuel you’ll need, and the bigger engines you’ll need. Having compared the height of the Atlas and the Saturn rockets, the Saturn rocket appears to be much taller (~48.8m and ~110.6m respectively), it doesn’t really make sense to me as to why the smaller rocket went further. Could someone explain how/why it works?
The first thing to keep in mind is that once you're moving in space, you keep moving in the same direction unless a force acts to change your trajectory. So distance, in itself, is not a factor.
What is a major factor is gravity. Both the Apollo lunar missions and the outer planet/extrasolar missions like Voyager (Titan IIIE) or New Horizons (Atlas) had to escape Earth's gravity. The outer planet missions had to build up enough speed to escape the sun as well, but the Apollo missions had to:
- brake into lunar orbit instead of flying right past;
- slow the orbiting lunar module further, to drop it to the moon's surface;
- brake while descending to the moon to land the LM softly;
- lift the LM's ascent stage against the moon's gravity, back to lunar orbit to rejoin the command/service module;
- boost out of the moon's orbit to head back to the Earth.
All of that takes a great deal of fuel, and instead of doing it with spacecraft massing less than one ton (478 kg New Horizons, 825 kg Voyager), it had to be done with spacecraft large enough to keep a crew alive, carry their experiments and moon rocks back and forth, and so on. The LM ascent stage alone was over two tons dry; the total mass of the command/service module and LM sent to the moon was over 40 tons.