Q: Why do basepairs pair up?
A: The 3 basepairs AU, GU and GC each consists of one short base and one long base. The bases in a basepair is bound together by hydrogen bonds.
Hydrogen-bonds are typically weak attractions between two different molecules that holds those two molecules close together. (Definition Brightstorm.com) For a hydrogen bond to form, bases need to have a certain distance to each other.
The short bases consists of a single ringed base (green C and blue U) and are called pyrimidines.
The long bases are double ringed and are called purines (yellow A and red G)
I found the picture on this page.
To remember I chose to take the first letters in the short and long bases and connect them to something that I think I can remember. If it works, it works. 
CUT - cut short
AG - A giant
Back to the pairing of bases. Picture that RNA folds up in helices like DNA. If you have two double ringed bases opposite each other, they will bump into each other and be too close to form proper hydrogen bonds. Two single rings would be too short to reach each other and thus not form bonds.
However a single ring and a double ring opposite each other have the perfect distance. It is the same in DNA, a short base opposite a longer base.
There is another reason that two long or short bases opposite each other won’t work, not thinking in the hydrogen bonds. The helix would be quite bumpy if basepairs of two short bases were next to a basepair of two long. It would screw with the biological machinery that have read and work along the line of basepairs in the helix of DNA.