Why do Mormons put so much more emphasis on the family than most other denominations do? What about their doctrine causes this? Or is it more because of culture/tradition?
I'm answering this only because of the year I spent in a Mormon (LDS) church. If anyone who is LDS would like to post a different answer, please do. This is based only on what I observed and learned during that time.
There are actually a few reasons for the emphasis on family. The first section below is presented to add emphasis to the "meat" of the answer found later on.
The first has to do with the fact that LDS doctrine teaches that you have to add works on top of the free salvation presented by Jesus' sacrifice if you want to attain to the highest level of Heaven (also referred to as the highest degree of glory.
This underlies a lot of why members of the LDS Church are such stellar examples of how to behave, or at least why so much emphasis is put on "being good". Doing the will of God and actually following the commandments is seen as necessary to achieve the highest level of glory. One of the basic principles of the LDS Church is that other denominations are false, because Christianity went into full apostasy. They view the concept of salvation by grace through faith apart from works as a false doctrine that encourages people to continue sinning while believing they are saved. I can't tell you how many times I heard Matthew 15:8 quoted in Sunday Schools and in services.
The emphasis on being a good person springs from this belief. This is, from what I've experienced, the most crucial aspect of LDS doctrine, and the belief that good works are necessary to future glory underlies a great deal of the emphasis on goodness. Part of the emphasis on family springs from this. Quite obviously, it is good to love your family and to have a close family.
Their teachings also include the idea that your family life continues on in heaven, and they want to be able to spend eternity with their families. This feeds the emphasis on family, because close families are less likely to drift away from the Church. Since the view is that a person who leaves the Church is going to attain a lower level of glory, they want very much to maintain close ties to keep their children and loved ones in the fold. The best way to do this is to encourage family time, and closeness in the family.
You can read the official LDS statement on the importance of family while a more "evangelical" and less "doctrinal" explanation can be found here.
I think this article from 5/2003 explains the LDS position quite clearly, as to why they believe in the importance of family ties. Family members that have died before the could be brought within the Mormon fold can also be given the saving ordinances by having someone stand in for them. This Mormon missionary prep guide may help explain this.
By stressing close family ties it helps to bind the religious community together, and serves as an example of how being Mormon can help society as a whole. There is also more pressure on trying to stay together because the families will try to convince couples to persevere, all to keep the family together.
It also makes is harder to leave the Mormon faith because these close relationships will be at risk, so when people choose to leave they will leave everything they hold dear behind them.
This story was interesting, I don't know how legit it is, but it can give some idea about what happens when people leave the Mormon church. I am not trying to attack the Mormon faith by this last link, but to some idea as to how family will work to try to keep people in the faith. If more religions cared as much about close family ties I doubt we would have so many divorces, in the US.
Faithful members of the LDS church have the opportunity to be married in the temple in a ceremony called a temple sealing. This sealing is one of the most sacred ordinances performed in the temple.
When a couple is sealed and married they don't hear the usual, "Until death do you part". This union doesn't end with death, rather it continues on into the eternities.
When a couple that has been previously sealed has children the blessing of the sealing extends to their children, as long as the parents have kept their covenants.
If a couple gets sealed after having children, the children enter the temple and the parents are sealed as husband and wife, and the children are sealed to their parents.
This sealing ordinance enables family ties to continue beyond death. This is why the family is a strong focus in the LDS church. The people in your family were likely close to you before you came to this earth, and if we keep our covenants, then we never have to lose them.
I know that this was answered several months ago but I would like to contribute my view on this since I am actually of the LDS Faith have left the LDS church at one point and returned after a lengthy period of time.
The family is the most central unit of the LDS church. All doctrine in the church revolves around the family unit. Without the saving ordnance of being Sealed in the temple to your husband/wife a member of the LDS church believes that they cannot attain the highest degree of glory. As David said there is a large emphasis on works as well as the saving grace of Christ. So beyond just getting sealed the couple and later their family must continue to work at being a Celestial family (the highest degree of glory according to LDS doctrine). Because of this LDS families that actually live the teachings of the gospel do place a heavy emphasis on close family relationships.
Now to clarify what @James Black said. As one who has belonged to the LDS church for most of my life and having left the church at one point as well. I can honestly say that I never felt like I was discriminated against by those who knew me or my family. In fact I have a sibling who is not currently part of the church and my family not only loves her still, but does not pressure her to come back to the church. Would we like her too? Of course we would, but part of the reason that LDS families that live the gospel are so strong is because the LDS church also puts such a heavy emphasis on accepting people for who they are, and not who we think they should be.