I found out that I cannot go up stairs backwards well at all and automatically turn my foot outward. What specific muscle is weak for this upstairs difficulty to occur, and why does turning the foot out help. Trying to figure out exercises to correct this.
I’ve tried walking up hills backwards, and I’ve seen other hikers doing it too.
It much harder than one would expect, and definitely exercises muscles that you didn’t even know you had.
I presume that simply continuing to do this, a little bit at a time, is the very exercise you need to do, to condition those muscles a little better. (I don’t know about walking up stairs backwards specifically, as distinct from walking up a hill.)
This is a complete WAG, but maybe it’s because, if you held your foot straight forward, the front part of your foot wouldn’t have anything to rest on, and it would feel precarious. When you go up stairs normally, you’re putting your weight on the front part of your foot, and it’s the back part that’s unsupported.
Does your body only react this way when walking backward up stairs, or – as @Senegoid mentions – does it behave the same way if you walked backward up a hill (ie, flat but not level) ?
How about on a flat and level surface ? Does walking backward produce the same result ?
Note: be really careful doing any/all of this. It needed to be said 
I think you are correct, it has something to do with the foot, or at least starting with the foot and working up the leg from there. The change to stepping on your heel instead of the ball of your is the biggest change in the process. Then in addition, you are likely unused to balancing in that mode, probably leaning back more than necessary, and there will be some unaccustomed strain on every muscle in your legs as a result.
Of course I got up and tried it. I had no problems.
part of it is that walking up stairs backwards is a skill to be developed. You are using different muscles differently than normal, your balance keeping is less automatic and sure and unthinking than when moving forward etc.
When I have to do it, I tend to try to step up onto the balls of my feet, that seems to help with balance, muscle strain etc. However I couldn’t actually tell you why turning your foot helps except that maybe somehow it helps with balance somehow.
Same here. Keeping weight on my heel doesn’t seem to be an issue and feels natural enough to me. I do ice skate a little bit, so perhaps my sense of balance and shifting weight either forward or back a little bit on my feet has been developed this way. Turning my foot slightly outward (which I noticed I did) helps me to center my weight over the area between my midfoot and heels, it seems.
The biomechanics involved in the human gait cycle is quite complex and evolved primarily to provide efficient forward (and to a lesser degree, lateral) locomotion. It involves the firing of a number of muscles with tendon insertions into bones resulting in locking and unlocking a number of joints in the ankle and foot in sequence. The sequence in reverse needed for backward (and in this case, elevation onto a step) ambulation is not nearly as efficient (it’s not a typical motion needed for bipeds).
This animated video highlights some of the complexity involved:
If I could quote Kelly Bundy “What would chairs look like if our knees were on the back of our legs”?
When I was in HS, the gym teacher would make us walk stairs if we forgot our gym clothes. I found I could read while I was doing this, and I was less likely to stumble if I went backwards up and down the stairs.