Why did the Magi bring gold, frankincense, and myrrh?

Biblical Archaeological Society  online article about "why."  An excerpt from their longer article “The Magi’s Gifts—Tribute or Treatment?” in Biblical Archaeology Review, January/February 2012.

Gift of Frankincense

Frankincense comes from tree resins. Frankincense was once greatly valued throughout the Middle East from Rome to India. It was very expensive and has a wonderful fragrance.

Frankincense occurs fifteen times in the Bible. According to Leviticus and other passages, frankincense was used in the Temple. It was sprinkled on burnt sacrifices (Leviticus 2:1-2). Song of Solomon 3:6; 4:14 reveals it was also an ingredient in perfume, in other words, a smell of love. Thus, frankincense is an olfactory reminder that God wants what is pleasing.  Sin was thought of as something putrid and rotting. Forgiveness purifies us by making us acceptable to God (pleasant smelling, purified, worthy).  The smell also signals to our brains that we are in a sacred space.  The site and smell of the Magi's frankincense would have been a symbol of that the Christ child was pure and worthy of worship.

Gift of Myrrh

Myrrh is extruded from the gum of the Commiphora myrrh plant. Myrrh was less expensive than frankincense, but was still highly valued.

Myrrh is referred to seventeen times in the Bible and is first mentioned in Genesis 37:25, where it was being carried by camels in a caravan. Myrrh was used for a variety of purposes in biblical times as a perfume (Song of Solomon 1:13; 3:6; 4:6, 14; 5:1, 5, 13), an anesthetic, for burial embalming (John 19:39), as an ingredient in anointing oil (Exodus 30:23-25), and to deodorize clothes. According to Esther 2:12, it was also used in a cosmetic for women. John 19:39 records that myrrh was used in Jesus’ burial.

The Magi's myrrh would have been a pleasing smell -- signifying that the baby was worthy of honor. Christians also see it as a reminder that this Christ child would one day grow up and die for our sins.

Source

The gift of Gold

This gift needs little explanation. Gold is what people gave in "tribute" to their king. A king's crown or other symbols of power were often made of gold. It was also used as money. More importantly, it was considered "pure" such that it was used in the creation of sacred objects, such as candlesticks and offering plates. The Bible has many interesting ways of thinking about gold, including that faith and the commandments were more precious than gold! 

Undoubtedly, the Magi brought gold to the baby as a way of honoring him as a king.

See this entry in our Advent "Magi" Art Workshop forum that connects the three gifts of the Magi to the three things God asks for in Micah 6:8.