From a legal perspective, the 1st amendment is indeed the primary driver ("Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."). The Supreme Court has long held that the government may not take any action that endorses a specific religious belief based on that amendment. Since the 10 commandments are explicitly supporting a religious purpose, they are almost always found to be in violation of the constitution (some courts have allowed the 10 commandments, but usually because of tradition of an old monument to avoid the inherent religious issue).
Conceptually, the reason for this separation by the founding fathers was to avoid religion getting entangled into government matters and vice versa. Thomas Jefferson expanded on this understanding with his letter to the Danbury Baptist Association in 1802:
Another important thing that the founder fathers understood that many people today fail to understand is the concept of "tyranny of the majority". The founding fathers understood this (James Madison wrote about it in a letter to Jefferson) since many people who originally came to the US (under their own will) did so to avoid the majority rule mentality (especially around religion) that prevented them from worshiping as they saw fit, living their lives as they saw fit, or put undue pressure on them just because they weren't in the majority faith or a majority group.Gentlemen
The affectionate sentiments of esteem & approbation which you are so good as to express towards me, on behalf of the Danbury Baptist association, give me the highest satisfaction. my duties dictate a faithful & zealous pursuit of the interests of my constituents, and in proportion as they are persuaded of my fidelity to those duties, the discharge of them becomes more & more pleasing.
Believing with you that religion is a matter which lies solely between man & his god, that he owes account to none other for his faith or his worship, that the legitimate powers of government reach actions only, and not opinions, I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should make no law respecting an establishment of religion, or prohibiting the free exercise thereof, thus building a wall of separation between church and state. [Congress thus inhibited from acts respecting religion, and the Executive authorised only to execute their acts, I have refrained from presenting even occasional performances of devotion presented indeed legally where an Executive is the legal head of a national church, but subject here, as religious exercises only to the voluntary regulations and discipline of each respective sect.] Adhering to this expression of the supreme will of the nation in behalf of the rights of conscience, I shall see with sincere satisfaction the progress of those sentiments which tend to restore to man all his natural rights, convinced he has no natural right in opposition to his social duties.
I reciprocate your kind prayers for the protection and blessing of the common Father and creator of man, and tender you for yourselves and your religious association, assurances of my high respect & esteem.
(signed)
Th Jeffersone
Jan.1.1802.
Keep in mind that you are free to put up a 10 commandments monument on your land if you wish. The government, however, can not do such things since they are required to support all citizens of the country, not just a majority.