Why can we not walk forward in space with no surface?

You can walk because you have friction with a surface that allows you to, basically,....continuously stop your foot and fall with your entire body forward. In space you cannot have this effect, a little bit like walking on ice!

Well, simply put, because there is no surface in the middle of space! Even with a surface though, we need to induce a reactive force to propel us forward.

Walking in general
To walk, we normally need a surface to provide us with friction that "reacts" to our footsteps. As we walk, our foot pushes diagonally at the ground, and the frictional force pushes back according to Newton's Third Law of Motion, and we move forward. The Normal force also reacts, allowing us to not fall through the ground. In the end, without a surface to walk on, we can't walk.

Walking on a surface in space
Furthermore, without the optimal amount of gravity, since friction depends on the Normal force (#F_"s/k" = mu_"s/k"F_N#), and the magnitude of the Normal force is directly proportional to the force due to gravity (#F_N prop F_g#), friction will be small even if you are standing on a surface.

Small amounts of friction don't help us as much to walk, so on the moon, due to the low gravity, you find it easier to just hop, pushing down at the moon's surface, causing the Normal force to momentarily increase beneath you, horizontal friction to increase, and your moving upward and forward to be easier.