The hand-waving explanation: The slower you're going, the easier (cheaper) it is to change direction. And for eliptical orbits, the outer-most part of the orbit is where you're going slow.
So to make a drastic change in direction (aka, a very different orbit):
1. First burn to move far away from the center of the orbit so that you're going very slow.
2. Then burn to make a large change in direction (orbit).
3. Then wait until you cross your desired final orbit, and burn again to close it.
The tradeoff is that these types of orbit change are very slow (because you want to be going very slow for the middle burn, which means you take ages to get there).
So to make a drastic change in direction (aka, a very different orbit):
1. First burn to move far away from the center of the orbit so that you're going very slow.
2. Then burn to make a large change in direction (orbit).
3. Then wait until you cross your desired final orbit, and burn again to close it.
The tradeoff is that these types of orbit change are very slow (because you want to be going very slow for the middle burn, which means you take ages to get there).