There may well be historical component to this, and I'd love to hear it if so, but speaking as an instructor, I can suggest a very simple answer: multiple choice is much quicker to grade. In any sort of classroom where you're doing a lot of grading - daily quizzes in a language class, say, or midterms in a 300-person lecture - multiple choice saves you a ton of time, and eliminates grader subjectivity to boot. (That's obviously less true for fill-in-the-blank questions, but even then you have to contend with misspellings, bad handwriting, and the like.)

From a historical perspective, it would be interesting to think about the implications of a shift from a recall-based model of knowledge to a recognition-based model.

It's also worth noting that machine-readable multiple choice formats, e.g. Scantron, have been available since the 1970s. So there might be a technologically-motivated shift as well.