Not a historian but I have a degree in Library and Information Science. Microfilm is just the middle of a continuum of micro/digital transfer that has been doing on for years. In one library I worked in we used to have "Microcards" which are from the 1940s and 50s. We have one machine left that could read them, but they are essentially just a piece of cardboard with the documents (in this case science articles) printed onto them in a very small but magnifiable format. After microcards came Microfishe. The problem with the cards was that even today it is very difficult to make a copy from them. With fische you could throw it onto the machine, locate the pages you wanted, and print them page by page. Problem is fische is kinda flimsy and can get damage/ lost easily. Fische is essentially just small pages of plastic, similar to the cards, with the benefit being it can be easily copied. However, if you loose a sheet its gone, and can be rather pricy to order a new copy.
After fische came microfilm. This was really popular around the 1970s. The benefit of microfilming was the it was study, copyable and compact. One of my friends who works in an archive still insists on microfilming, because of its beneficial storage properties. Microfilm made in the 1970s is still accessible today. You can take a microfilm from 1970, put it into the machine and its still readable.
Prior to microfilm documents would have been preserved on cards or fische, but after microfilm came out these two methods become obsolete. The reality is, in most research libraries that existed from 1940s onward you can probably still find large collections of fische (less with cards, although I knew a scientist who was reduced to borrowing a microcard and reading it with a microscope and copying the information by hand).
You might wonder why not digitize everything...well it comes down to a couple of reasons
I can't say for sure why microfilming took off when it did, but I suspect a lot of it has to do with cost savings. In Canada a lot of the original documents were microfilmed and then the originals were discarded, as the information in them now existed in a smaller, accessible and more easily copyable format. The danger in this is that if anything went wrong in the microfilming process (missed a page, it turned out fuzzy) you can't go back and redo it. But in the 1970s this must have seemed like the best of both worlds, you could keep the information, make it easily accessible, and stop having to pay for storage of all these musty old books and documents.
When digitization came along in some cases all they could do was digitize the microfilm. Canadian genealogists run into this all the time and the census, land registration and ship manifest lists exist only in this format. On the other hand, at least there is SOMETHING left to access.
If I am remembering correctly it is estimated that if you store microfilm properly it should be readable in 200 years. For a librarian you can put it on a shelf and forget about it. Compare this to digitized material which need to constantly be migrated. Its easy to see why some of us still see microfilm as a very useful resource.
Sorry this was so long and rambling.
Quite a few reasons: