"Incense" (or frankincense) was a costly substance that created a fragrant smoke when burned. As the verse indicates, in this case it symbolizes our prayers rising to God.
The phrase, "a Deity nigh" refers to the Christ Child himself, who was "near" from the perspective of the three Kings who had traveled to worship him and offer Him gifts. Nigh is an old word for near. And Deity means divine, or God.
"Owns" probably does not mean the plain fact that one of the kings had given Jesus the gift of some frankincense. While that is true, it probably refers to the imagery of the smoke rising like our prayers to God. Our prayers belong to God in the sense that prayer should be offered only to God. And this Christ child was God come in the flesh!
One can picture the scene with Joseph, Mary, the baby Jesus, and the wise men, or Magi as they were called.
To paraphrase the line, "Incense--representing our prayers--belongs to this child before us, who is God in the flesh.
Frankincense to offer have I;
Incense owns a deity nigh;
Prayer and praising, voices raising,
Worshipping God on high.