What other explanations exist for this prophecy? What do other denominations think it means that Elijah will "turn the hearts of fathers to their children and the hearts of children to their fathers?"
The Bible teaches that the Spirit of Elijah would be sent to the earth a total of 5 times (5 is the number of grace1). By Spirit of Elijah, I mean God's spirit anointing a man to bring a message, in the same way that the first Elijah was anointed (a straight shooter, hard hitter, come out of the wilderness, etc).
First, Elijah Himself
An examination of Elijah's traits may be beyond the scope of this question. If requested, I can revisit later.
Second - Elisha
Third - John the Baptist (Forerunner to Jesus' First Coming)
Fourth - Forerunner to Jesus' Second Coming
Reading Malachi 3 as referring to John the Baptist's fore-running of Jesus' first coming, Malachi 4 can be read as another fore-runner preparing the way for Jesus' second coming. This man will turn the hearts of the children (the believer's of today) to the original faith preached by the early church apostles (fathers), just as John turned the hearts of the fathers of Israel to the "children" that received his message of the coming Christ. Malachi 4:5 predicts that this messenger will come before the great and dreadful day of the Lord, and if his message is rejected, the earth will be smitten with a curse.
Luke 1:17 specifically states that John would turn the hearts of the fathers to the children (Jesus' first coming), but does not include the Malachi 4:5 reference to turning the hearts of the children to the fathers (Jesus' second coming). This is similar to how Jesus, in the synagogue, began to read from Isaiah 61:1-2, but stopped halfway through verse 2 before he got to "and the day of vengeance of our God". He would fulfill that part of that verse at his second coming, but not his first.
Fifth - One of the Two Witnesses to Israel
See my response to this related question: Who Are the Two Witnesses?
To briefly summarize that more detailed response, the characteristics of the two witnesses are so similar to Moses and Elijah that we can conclude that the anointing of the two witnesses will be the same that was with Moses and Elijah. This is the fifth and final time that the spirit and power of Elijah will be manifest on earth.
1 See EW Bullinger's Number in Scripture, on the number five.
You ask for what groups besides the LDS have alternate views on the meaning of this passage. There is group knows as Branhamists (or by their own internal name of Message Believers) who, like me, believe that the fourth Elijah mentioned in Malachi came to earth in the person of William Marrion Branham. Like the LDS, this is a reconstructionist faith and we believe the original apostolic faith of the Bible has been lost and a new prophet sent to restore it.
The return ministry of Elijah is according to the Bible. The ordained one will come in the Spirit of Elijah doing the same works of Elijah: