How does moving your legs forward on a swing propel it forward instead of backward? Moving your weight back is equivalent to moving the pivot forward, like having a ball on a string and moving your hand forward, then, when it moves forward, moving your hand back.
What I don't understand is how you can propel it forward by only moving your legs forward, which should propel it back.
 A: You are not actually propelling it forward as such. What you are doing is that each time you swing you  raise your legs at the top of each swing. This moves your bodies centre-of-mass to a greater height creating more potential energy. This potential energy is in turn transferred into kinetic energy, meaning after each swing you will go faster and higher and so on. EDIT Added: Note that you are swinging your feet in phase with the swing to make it go higher, making this a resonance effect. (added edit thanks to David white' comment below - thanks David)
A: It seems counterintuitive, somebody moving a swing forward by only swinging their legs forward: If somebody is standing on a frictionless base on a level surface and they thrust something forward without letting go of it, the combined centre of gravity remains the same. Mass times displacement is the same. But, when one object is above the other on a pivoting lever, the object furthest from the fulcrum has the most leverage, so, if they are the same size, the object nearer the fulcrum will move further back, shifting the centre of gravity back.
Regardless of the size, the mass times displacement is greater for the object nearest the fulcrum. So when somebody on a swing swings their legs forward, because their legs are below the rest of their body, they actually shift their centre of gravity back slightly. It is the same as somebody about to fall over a cliff leaning over it to regain their balance.
Because their legs rotate forward and/or body rotates back, there is also a moment of rotational inertia which rotates the swing back slightly more, by trying to push the pivots forward as you press on the chains. When you lean back and swing your legs forward, everything except the bit on the seat rotates backwards, giving you maximum  forward propulsion, both from rotational inertia, and shift of centre of gravity which is equivalent to moving the pivot forward.

