Let me preface this by stating that I learned the trick for dividing fractions, simply multiplying one of the terms by the other's inverse, a long time ago. However I always find things more understandable when I have a good concrete example of how something works. My girlfriend is very strong in math, and wants to be an elementary or middle school math teacher, and she gave me the following example to think about. Say for example you have 4 quarts of ice cream, and you want to split it up into cups that each hold 2/3 of a quart. First you take 2/3 of each quart of ice cream, leaving 1/3 quarts in each ice cream container. Since you started with the 4, you have now filled up 4 cups. You have four 1/3 quart portions left and you want to fill up 2/3 quart cups with that portion, so two of those 1/3 quart portions will fill each cup, meaning two more cups are needed.

I thought it was a very good way to think about division. Perhaps I was taught an example like that in school, but all I remember was the simple inversion trick. I didn't get a really good understanding for what I was doing.

Another thing that helps with fractions is to change what I subvocalize when I see one. I was taught to think "two thirds" when I see 2/3 for example. If on the other hand I think "out of three parts, take two", when I see 2/3, I get a much better understanding for what the fraction actually means.