Burglary has been a recognized English word for at least 800 years. It means to break into a house at night with the intent to commit a felony. The word burglar has been with us just as long.
Burgle is a silly back-formation playing on these words. It was coined in 1869, and was used only humorously or erroneously; it was clearly not a real word. Its coinage may have been influenced by the barbaric Americanism of "burglarize", a ridiculous invention that appeared c. 1865 (the first appearances at that time are mocking those foolish North Americans for using such a word).
However, it was true that English lacked an good verb for "to sneak around stealing stuff", unless you counted pinch, filch, crib, and, oh yeah, steal (swipe and nick, used in this sense, didn't appear until after burgle), so burgle caught on, and can now be found in most modern dictionaries.
It is interesting to note that while burgle is a new word of no good pedigree, other weird variants have been around for centuries; for example, for a time 'burglar' was in competition with the awkward alternate form burglarer (in use since the late 1500s); the adverb burglarly was formed after the same manner as feloniously (also from c. 1500); and the tongue-twister adjectival form burglarious appeared late on the scene (1769) with no apparent controversy.