How are neon sign transformers built?
Ordinary power transformers are designed to have good voltage regulation (the ratio of input to output voltage is fairly constant regardless of load)- the flux is mostly linked between primary and secondary and the leakage inductance is minimized. 

Neon sign transformers are made to have high leakage inductance- ferromagnetic "shunts" are inserted (magnetically) in parallel with the secondary, which is similar electrically to having a large inductor in series with the output. That makes the transformer more of an AC constant current source than a constant voltage source. It allows the voltage across the neon tube to rise in order to start the ionization, and prevents excessive current from being drawn once the discharge starts. Below is a photo of a disassembled Neon Sign Transformer from this website (arrows and caption added).  



It's possible to fool with the shunts in order to change this behavior (removing them increases the short-circuit current). 

Modern neon sign "transformers" are typically switching power supplies designed to have similar characteristics to the older style of mains-frequency transformer. The old style ones are practically indestructible, and the new ones are not going to last nearly as long.