The story is that St. Francis' mother, Lady Pica, was finding her labour to be particularly difficult. Her husband was abroad in France on one of his business trips, and she was at home with only her household staff in attendance. Whether it was her idea, or another's, the thought came that perhaps a walk would be good, and so she set out a short distance to the nearby stable. Apparently it did the trick. The baby hastened his entrance into the world, and ended up being born in the stable.
Whether this is historically accurate, or is simply a pious legend, we can never know for certain. After the death of St. Francis there was an effort (done with the best of motives) to look at his life and to see a certain conformity with the life of Christ. Of course, many aspects of St. Francis' life were very much like those of our Lord's life - his poverty, his preaching of divine love, his receiving of the stigmata. The list could go on.
So, does this story reflect the reality of his birth? Or does it, perhaps, represent a pious attempt to make the saint's life conform to that of Christ in every detail? I don't know for certain, although I tend to believe the tradition attached to this oratory. It would be just like God to work things out like that.
On one of our pilgrimages we offered Mass in San Francesco Piccolino because we were few enough to fit inside. I remember thinking, "If this isn't the place, it should be."