Why do Catholics worship Mary?

The standard understanding of worship is honor or reverence shown to a deity. Catholics do not see Mary as a deity. They do understand that Mary is worthy of great honor.

Mary is the mother of Jesus. Jesus acknowledged His title of "King". (1) According to Jewish tradition, this makes Mary the "Queen Mother" or "Gebirah". "Gebirah" or "Queen Mother" is a special role with significant power. When Bathsheba went to her son King Solomon on the behalf of Adonijah, King Solomon said, “Make your request, my mother, for I will not refuse you.” (2).

Look then to the Wedding Feast of Cana, when Mary went to Jesus on behalf of the wedding guests. Jesus said, “My hour has not yet come” yet Mary confidently orders the servants to do what Jesus says and Jesus did not refuse her. (3)

Catholics rightfully know Jesus is the "King of Heaven and Earth" and Jesus is "King of Kings" (4). This makes Mary the "Queen of Heaven and Earth" and "Queen of Queens". Mary is the reflection of the divinity of Jesus, but not divine. She is owed great honor and veneration as "Queen and Mother of All Believers".

Citations: (1) John 18:36, (2) 1 Kings 2:19-20, (3) John 2:2-10, (4) Revelation 19:16