“an angel of the Lord appeared to him, standing.” The angel did not walk in, but materialized right where he stood, going from invisible to visible on the right side of the altar.
“on the right side of the altar of incense.” The right side (or right hand) was the side of blessing, something that shows up a number of times in Scripture (cp. Matt. 25:33; Prov. 3:16). In this case, the Temple faced east, so the right side of the altar of incense was the south side. In the Temple, God was the priority and He faced east, so the “side” that something was on was determined by Him, not by the worshipper looking west at Him. To God looking east, the angel was on the right side of the altar of incense, while to Zechariah looking west, the angel was on its left side. That looking east is the priority also shows up in descriptions of the Temple in books such as Psalms and Ezekiel (cp. Ps. 110:1; Ezek. 40-48).