In LDS belief, why are eternal families so important?

I don't know if we'll ever truly understand the importance of eternal families until we get into the afterlife, but I think I can provide at least one possible explanation.

One of our basic beliefs is that we are all (literally) children of God. We were placed here on this earth to learn and grow to become exalted, which is to be like our Father in Heaven, in the same sense a child becomes like his or her parents. One of the other things we know is that Heavenly Father is married, and that with Her, they created all of us (Their children).

If our ultimate purpose of this life is to learn what is necessary to become like our Heavenly Father, then having an eternal family like Him would be within reason.

I know it's not a great answer, but it's a start.

The significance of eternal families is not to organize new lineages, it is to seal the lineage of God together eternally.

In the end there will be two ultimate outcomes to your life on Earth: either you will return to live with our Heavenly Father, or you will not. Those who do not return to the Father will receive varying degrees of glory in the telestial or terrestrial kingdoms, or they will be cast out into outer darkness. Those who do return to the Father will receive celestial glory, exaltation, and eternal life.

In order to receive celestial glory, you must be sealed in the temple. This sealing is more profound that simply being sealed to you parents, your wife and children, because each of your family members is in turn sealed to their spouses and children, and your spouses are sealed to their parents. Ultimately, we will ALL be sealed together, and everyone will be sealed to Christ. This is the significance of eternal families: so that we can all be sealed to Christ and our Father in Heaven. In the Celestial Kingdom we will all be sealed together as one eternal family. It's all part of the plan of happiness, which was presented in the premortal realm.

IN THE PREMORTAL REALM, spirit sons and daughters knew and worshipped God as their Eternal Father and accepted His plan by which His children could obtain a physical body and gain earthly experience to progress toward perfection and ultimately realize their divine destiny as heirs of eternal life. The divine plan of happiness enables family relationships to be perpetuated beyond the grave. Sacred ordinances and covenants available in holy temples make it possible for individuals to return to the presence of God and for families to be united eternally. (The Family: A Proclomation to the World)

The proclomation specifies families, but ultimately we are all one family.

This is why Temple work is so important, everyone must be accounted for, everyone must be given the opportunity to be sealed to Christ. This is also why the Millennium when Christ returns is going to last 1,000 years, so that temple work can be completed for every spirit child who came to earth to receive a body, and so that they can be given an opportunity to choose to accept those blessings.

Eternal families are based in a life in that same organization after the Resurrection and Judgement (cf. Doc & Cov 130:2).

Regarding the resurrection,

As part of His eternal plan, God has provided a resurrection for everyone.

1. Everyone who has lived will be resurrected (see 1 Corinthians 15:21–22; Alma 11:41; D&C 29:26; 2 Nephi 9:22).
2. Resurrection is the reuniting of the physical body and the spirit after death (see D&C 88:14–17; Alma 11:43; 40:23; 2 Nephi 9:12).
3. We cannot receive a fulness of joy when the spirit and the body are separated (see D&C 93:33–34; 45:17; 138:50).
4. Resurrection is brought about by the power of God (see John 5:21; Acts 26:8; 1 Corinthians 6:14; 2 Nephi 9:12).
5. Resurrection restores every limb and joint to its proper and perfect frame (see Alma 11:43–44; 40:23; 41:2).

Doctrines of the Gospel Student Manual; Chapter 32: The Resurrection and the Judgment

While the eternal intelligences inhabit the spirits created by our Eternal Parents (cf. Doc & Cov 93:29 and Abraham 3:21-22), our spirits will, in turn, inhabit the physical bodies which were created for us in this life by our earthly parents (cf. Doc & Cov 88:15).

Our sons and daughters will be organized in a grand family with Adam and Eve at the head of those from this earth. This grand family is all organized within the Eternal family that includes all of the spirit children of our heavenly parents—including those from the innumerable worlds which God has created. Both lineages are important as they signify the various stages of our eternal progression and through the estates which we have kept (cf. Abraham 3:26).

This lineage is not a "ranking" system of "best to worst," rather it is a system of organization based upon the basic family unit. Only those in the highest degree of the Celestial Kingdom will be a part of this grand family and are those who are "heirs of God, and joint-heirs with Christ"(Romans 8:17).