Question

What causes the Milky Way to move through space?

Answer 1

We still aren't so sure.

Explanation:

Much of the Universe we believe is still expanding outwards in one direction from what we can see from Earth today.

Most Galaxies are moving away from us, yet one Galaxy is not & it on a collision course with ours; The Andromeda Galaxy.

Don't worry, it's been estimated it wont hit Earth but we shouldn't worry about it because we wont see it in our life time, collide or come close enough that the night sky glows with it's many suns.

This isn't the first time either a galaxy had collide with another.
I recommend checking this link out at the Hubble Site to see some incredible pictures of colliding galaxies:
http://hubblesite.org/gallery/album/galaxy/interacting/

Before I finish though.

I personally believe that the Universe is still expanding & the rapid movement of Galaxies outwards is the explosion sending us outwards, but possibly the nicest idea is that we may be riding on the explosion itself, sending us deep into the possibly never ending cosmos, our Galaxy it's self, a ship for us to ride upon & if there really isn't any other life out there & the Universe it's self is something of an 'entity', we may be the Universe's way of experiencing itself, so lets experience it.

Hope this helps.