Why do Latter-day Saints believe that the US Constitution is divinely inspired and what does that mean?

I'd like to supplement the excellent answer by depperm with a few other scriptures and some theological background to address some of the discussion raised in the comments.

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High-level overview on beliefs about the US Constitution

There are 5 verses in the Doctrine & Covenants that form the principal, direct, scriptural background to Latter-day Saint beliefs about the Constitution:

5 And that law of the land which is constitutional, supporting that principle of freedom in maintaining rights and privileges, belongs to all mankind, and is justifiable before me.

6 Therefore, I, the Lord, justify you, and your brethren of my church, in befriending that law which is the constitutional law of the land; (Doctrine & Covenants 98:5-6)

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77 According to the laws and constitution of the people, which I have suffered to be established, and should be maintained for the rights and protection of all flesh, according to just and holy principles;

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80 And for this purpose have I established the Constitution of this land, by the hands of wise men whom I raised up unto this very purpose, and redeemed the land by the shedding of blood. (Doctrine & Covenants 101:77,80)

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54 Have mercy, O Lord, upon all the nations of the earth; have mercy upon the rulers of our land; may those principles, which were so honorably and nobly defended, namely, the Constitution of our land, by our fathers, be established forever. (Doctrine & Covenants 109:54)

The earliest of these statements was written in August of 1833.

I also highly recommend the April 2021 sermon/article Defending Our Divinely Inspired Constitution by Dallin H. Oaks, a man who has served as a University President, a Supreme Court Justice for the state of Utah and, more recently, as an apostle of The Church of Jesus Christ of Latter-day Saints.

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Inspiration & Stewardship

Sometimes Christians hear that Latter-day Saints believe the US Constitution is Divinely Inspired and come to the conclusion that this means Latter-day Saints believe the US Constitution should be part of the scriptural canon. This is incorrect, but it is an understandable misunderstanding.

Latter-day Saints believe that God inspires people all over the world, and reveals truth to all who will sincerely & faithfully seek it (see Moroni 10:3-7). This means many thoughts, lessons, sermons, phone calls, and other day-to-day promptings are inspired, because God brought people knowledge, understanding, and insight, but it does not mean that all of it belongs in the scriptural canon.

One of the key principals of leadership in The Church of Jesus Christ of Latter-day Saints is the concept of stewardship. This principal certainly has a Biblical basis (e.g. Gal. 2:7), and includes the belief that when God gives people a responsibility for a portion of the flock, they are entitled to revelation for that portion of the flock.

A Bishop can receive revelation for his ward. A Stake President (roughly comparable to a Catholic archbishop) for his stake. A Relief Society President or a Young Women President for those they oversee. Parents for their family. The prophet for the world. However, the Bishop is not entitled to receive revelation for the Stake President, and so on.

God is not going to reveal conflicting eternal truths to different people, but He may well give different guidance on handling a specific situation to different people in different circumstances.

If God raised up the founding fathers of the United States for the purpose of creating a free country and writing & implementing the US Constitution, then He certainly could (and I believe He did) inspire them in their frail, imperfect, human efforts to develop a system of government rooted in the belief that "all men are created equal, that they are endowed by their Creator with certain unalienable Rights".

This means God provided enlightenment, direction, and insight to the people He had appointed to this role and responsibility in life. It does not mean the things they wrote are intended to give commandments or reveal the plan of salvation to all the world.

Although Latter-day Saint leaders speak regularly, including in global, general conferences, the process of canonization is usually slow, and material is only added to the scriptural canon with careful consideration and unanimous approval of the leaders of the church.

A discussion of Latter-day Saint beliefs about canon can be found here, including the statement:

from time to time the legal authorities of the Church will see fit to formally add to the collection of scripture.

The last time the scriptural canon of The Church of Jesus Christ of Latter-day Saints was updated was Official Declaration 2, sustained in 1978 and added to the Doctrine & Covenants in 1981 (source).

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Conclusion

Latter-day Saints believe God inspired the founding fathers of the United States, including in their drafting of the US Constitution. Latter-day Saints do not believe the Constitution is inerrant, the product of plenary verbal inspiration, or canonical scripture.

Revelation received by God's authorized representative speaking to the world ("the prophet", "prophetic revelation") is seen as generally applicable and authoritative to all. I believe God is just as capable of speaking authoritatively today as He did to Moses or Peter, and as such, the scriptural canon is open. Latter-day Saints should be open to anything God has to say.

We also believe that guidance may be given to individuals in specific circumstances for specific duties. God solves many problems this way, but this is not how He reveals the plan of salvation or gives commandments to the world.

Disclaimer--these thoughts are products of my own study and do not constitute official statements by The Church of Jesus Christ of Latter-day Saints