So I can speak to two legal historical features of the United States Constitution that are major contributors to the phenomenon you're describing.

First, the United States Constitution has, since the early 19th century, been viewed as positive law. Specifically, Article 6, Clause 2 of the Constitution provides that "[t]his Constitution and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding." This clause unambiguously states that the federal Constitution and laws override any contrary state constitutions and laws, but it was less immediately clear what the legal relationship was between the Constitution and federal laws made "in [p]ursuance thereof." The question was settled in 1803, less than thirty years after the adoption of the Constitution, when the United States Supreme Court decided Marbury v. Madison. The case established, among other things, that federal laws which were inconsistent with the Constitution were void, and that it was the Supreme Court who would ultimately decide the question of that consistency. The Supreme Court has further developed its views of Constitutional supremacy and judicial review since then, but from fairly early in the history of the United States the Constitution has been considered to be not only law, but law that overrides all contrary laws.

Second, the Constitution, by its terms, contemplates amendment, but the amendment process is challenging. Specifically, Article 5 of the Constitution requires that amendments be proposed by a two-thirds majority in the United States Congress, or through Constitutional convention when requested by two-thirds of the states. Once proposed, for the amendments to take effect, they must be ratified (in one of two manners) by three-quarters of the states. So a very significant supermajority is required for proposal and adoption of amendments to the Constitution, and consensus on that scale can be elusive. There have only been 33 formally proposed amendments, of which 27 have been ratified. Ten of those were adopted more or less immediately following the ratification of the Constitution itself. All such amendments have been proposed by Congress, there has never been an Article 5 Constitutional convention.

So a large part of the reason that the Constitution has a powerful influence on public policy and political debate in the United States is because it is quite literally the overriding law of the nation, and it is a law that is very challenging to change. There are other contributing factors to the veneration of the Constitution (and its authors) that are more ideological, but I'll leave that to other historians better equipped to address those points.

Sources:

The Constitution of the United States of America

Marbury v. Madison, 5 U.S. (1 Cranch) 137 (1803)

William W. Van Alstyne, A Critical Guide to Marbury V. Madison, 18 Duke Law Journal 1-47 (1969)

The Constitution of the United States of America: Analysis and Interpretation, Congressional Research Service (2013 Edition)

This might be a little bit more of a legal question, or at least a comparative politics question, but here's some thoughts.

If you are looking at the American political system from the UK, I think a major difference between the two is the concept of parliamentary sovereignty in the United Kingdom, which more or less means that parliament is the supreme lawmaker, that UK courts must uphold parliamentary acts, and that no parliament is bound by decisions of previous parliaments (not a British Constitutional lawyer, so please correct me where I'm wrong). There also aren't separation of powers in the UK - parliament appoints the Prime Minister and Cabinet, and determines the shape of the judicial system (the Supreme Court in the UK is all of 11 years old, and previous to this the highest court of appeals were the "Law Lords" in the House of Lords). Furthermore, the UK is an asymmetric federation, meaning that Scotland, Wales and Northern Ireland have legislatures with devolved powers specifically because the British parliament has passed legislation making it so (and it's asymmetric because England does not have an equivalent level of government).

This is very different from the American system. A major difference off the bat is that the Constitution, as amended, is the supreme law in the United States. On top of this, there is a much clearer separation of powers between the executive branch, the legislative branch, and the judicial branch in the US system, and essentially all branches have to defer to the Constitution. This is why, for example, the US Supreme Court has the power of judicial review over acts of Congress or executive decisions by the Presidency/Federal Government (even though judicial review isn't really an explicit power granted in the Constitution).

On top of these constitutional prescriptions for the federal government, as noted by the name, the US is very much a federation, meaning that per the Constitution, Congress has specific, enumerated lawmaking powers (Article I Section 8), as well as some leeway for legislating in areas related to these powers, noticeably through the Commerce Clause. However, as per Amendment X, specific powers not held by Congress revert to the states and the people (so, for example, this is why states determine educational curricula in the United States, because the federal government doesn't have explicit enumerated powers to legislate in this area).

A lot of the reasons for the US Constitution being written this way were in part a reaction to the British parliamentary system, but also through the influence of Enlightenment thinking (notably Montesquieu and his separation of powers, but also with influences from Scottish Enlightenment thinkers, most directly John Witherspoon). It tends to get underplayed today in the hagiography, but a lot of the US Constitutional order and its checks and balances between federal and state governments, and between branches of governments at each level, was the result of political compromise and expediency.

As for why there is such a particularly strong focus on the Constitution in the US - besides the reasons mentioned above, a major part of it has been specifically because of how US courts have evolved, and how their rulings tend to be interpreted. Because of fears of political backlash, there has been a tendency for Congress to essentially "punt" on lawmaking questions and defer to Supreme Court rulings. Since the Supreme Court can't legislate, but only pass rulings based on its interpretation of the Constitution, this tends to place particular emphasis on particular juridical and legal schools of thought (one rather well known but relatively recent one is the idea of "original intent", meaning that justices can only judge on the constitutionality of a law, for example, by interpreting the Constitution specifically as the writers thought they were writing it). That last definition is a vast oversimplification, but nevertheless - the Constitution is the supreme law of the United States, to the point of government employees and military personnel swearing an oath to uphold and defend it. It was meant to be the keystone of a system of federal subjects and separated branches of government.