In case you were wondering in particular how the AC circuit is just moving electrons back and forth, but there’s somehow electrons moving in a single direction in the DC part, look at the two circuits in si_blakely’s link. Look at where the top left horizontal line meets the square of diodes. In the top picture, the line is red, and it’s pulling electrons from the red diode. In the bottom picture, where it’s blue, it’s then pushing those electrons through the blue diode. So while in most of the AC line the same electrons just go back and forth, at the corner of the square, the line takes the electrons from one diode and gives them to the other. The same is true at the corner on the right. At the top and bottom corners, the electrons always come in the top corner, and leave the bottom corner.
There always has to be someone to ruin a perfectly good joke by giving an actual explanation. Hrmmmph!
My theory trumps your reality!!!
Phlogiston is the proper explanation!
Chill.
You know I stared at that picture for a while and tried to understand the contradiction I was seeing, but it wasn’t until your post that elucidated it that I think I’ve figured it out. Thank you for your help! Now I don’t have to worry about some huge build up or lack of electrons inside the converter as the DC side pulls them away (or pushes them inside). Phew! 
Coming back to your sloshing water analogy
Imagine a seawall - the waves push in and pull away. Now put big pools behind the seawall, with one way gates. When a wave comes in, it pushes water into the pool. But as the wave retreats, the gates close. Now the water level in the pool is higher than before, and you can tap the pool to get a continuous (but somewhat lumpy) flow. Some foam baffles floating on top of the water reduce the sloshing in the pool, and your output will be smooth and constant.
A second pool collects the output (after doing some work) of the first. This is also open to the sea, with one way outlet gates. When the waves are pushing, the gates are closed, and the pool just fills up from the inflow. As the wave retreats, the gates open and water in the pool flows out to the sea.
This is halfwave rectification - fullwave rectification is somewhat more complex (beyond the ability of this analogy), but I hope it helps.
Awesome stuff! That analogy really does help and I think I understand halfwave rectification pretty well now, and it definitely clears up the “contradiction” I was seeing/asking about.
Now I need to look into this fullwave rectification! 
That involves flipping the sea upside-down when the wave crosses the middle point. 