We get a lot of this type of question: "we know X, but can we scientifically prove X?" The answer, almost invariably in the field of strength training, is no.
Science is dead; long live science
There's an enormous number of variables in strength training: nutrition, hormones, time of year, mood, prior training, genetics, exercise selection, proper form, proper coaching, degree of coaching, rest, program interruptions, mobility...the list goes on. I have yet to see more than a handful of studies that control adequately for even a few of these. (To get an idea of some of these problems, see 2011 in Strength Training (PDF), a strength-science literature review by someone affiliated with Mark Rippetoe, who authored Starting Strength, which is very similar to StrongLifts.)
Another massive problem with any scientific evidence regarding strength training is the problem of the novice effect. To wit: if someone's been sitting on a couch for ten years, getting them to do any exercise at all will produce immediate and dramatic changes. From this fact springs a million advertising claims.
So a dozen untrained grandmothers can improve their eyebrow-muscle endurance by using a calf raise machine eight times a day while eating nothing but graham crackers. Great. But what would have happened if they had done squats, deadlifts and chin-ups? Or marathon training? Or if they did squats, did they do them correctly? According to which standard? (And why do we care about eyebrow-muscle endurance, anyway?) This kind of science, where variables are controlled and compared in a useful manner, is apparently hard to produce.
"Science" in sports, bodybuilding, fitness, and strength training is almost always of the N=1 variety. That is to say, people experiment on themselves. (Or the people they coach, making N>1, but N is still too heterogeneous to make it peer reviewable.) Kurtis Frank has an excellent observation on this point:
[Empiricism is one of the foundations from which science is built.] That is what most coaches and athletes who get results are doing. They run a trial, find a result (good or bad) and design future trials to try and isolate the variable. A temporal relation always exists in these trials, there is usually some modicum of control placed on the subjects (don’t do cardio on this powerlifting regimen, I want to see if it works without cardio fucking up your gains), and people are ‘on to something’ when their results are compared and contrasted against other person’s results at athletic conferences.
Its not indexed in medline, they don’t use nice laboratory methods or questionnaires, and they don’t write everything they do up and submit it for peer review. That being said though, it is empirical evidence gained by the scientific method (or some very similar beast) that is tested and retested to try and find ‘what’ works.
You want science about a lifting or fitness program? Ask somebody who did it personally. Ask a coach who had fifty high school football players do it. Do it yourself. Read up on basic science, paeans to the program, and critiques of the program.
StrongLifts is a lot like Starting Strength, and both of them are a lot like any other 3x5 or 5x5 novice barbell program. You squat, then next time you squat more weight. It works. Barbell lifting is not based on made-up mysticism like "muscle confusion" and it isn't an infomercial trying to get you to buy some overpriced coat hanger with a treadmill attachment. Lifting weights (like fixing your diet, running, and playing sports) is one of those things that works.
And once you're squatting, deadlifting and pressing heavy barbells, your adherence to the program--eating enough, resting enough, lifting the weight correctly--is way, way more important than the program you choose. It's not science, but it's true.
Oh, the scientific principle that the program is based on is supercompensation. That's why programs like StrongLifts and Starting Strength work. You lift, your body is stimulated to grow, you lift heavier. Repeat until it doesn't work anymore, at which point you're much stronger. For further reading, Rippetoe and Kilgore's book Practical Programming is a fine choice.