The stranger did not lodge in the street; i.e. "I did not suffer any stranger who came under my notice to lodge in the street, but, like Abraham (Genesis 18:2-8), went out to him, and invited him in, to partake of my hospitality."
This is still the practice of Arab sheikhs in Syria, Palestine, and the adjacent countries. But I opened my doors to the traveller; literally, to the way; i.e. "my house gave on the street, and I kept my house door open."
Compare the Mishna, "Let thy house be open to the street" ('Pirke Aboth,' § 5).