I majored in American History, and while I'm not a constitutional expert I'll share some background that might be helpful.

It's important to understand that the American national government at the time of the Constitutional Convention of 1787 was extremely weak. The Congress established by the Articles of Confederation that preceded the Federal Government established by the Constitution had scant limited powers. So limited, in fact, that they could not levy a tax on their own but had to request money form the states who could either comply or deny the Congress their request with little to no consequence. This of course stymied some national efforts, like the early American Army. In 1779, George Washington complained in a letter to John Jay that the money they had apportioned to sustain the war effort "will scarcely purchase a wagon load of provisions." President of the Congress John Jay responded by requesting $45 million from the states, none of whom complied.

The weakness of the national government became more apparent in the post-Revolutionary years (Shay's Rebellion being the most-cited example) and a lot of Americans were really wondering if the union of states could (or would, or should) last.

When the Constitutional Convention was gaveled into session in 1787, it's easy to see how a stronger national government was favorable to a lot of the delegates. Of course, the views on exactly how strong it should be was a spectrum (see the Federalist and Anti-Federalist papers if you want specific essays that are exemplars of this spectrum).

I don't want to give a pithy answer, but in short, they agreed that the Congress needed to have some powers, like be able to levy taxes, regulate commerce, declare war, etc.; basically all of the powers listed in Article One (including establish a Post Office!). These listed powers are the explicit powers of Congress granted by the Constitution. The final draft of the Constitution, of course, included a Bill of Rights that appeased some of those on the spectrum that wanted to ensure they weren't giving the national government too much power (Jefferson, for example, although fun fact: Jefferson was not at the Constitutional Convention at all, because he was busy being our Ambassador to France). Anyway, the 9th and 10th Amendments are pretty clear that Congress couldn't do EVERYTHING that it wanted. It would still be limited:

"The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people."

And

"The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people."

Essentially, Congress has a list of things in Article One that they can do, and these amendments say "but, everything else that's not explicitly in the jurisdiction of the national government remains in the jurisdiction of the states, or the people (and, by the way, there are other rights that we didn't list explicitly, and those remain with the states and the people too)."

I wouldn't say the Post Office was so important that it had to be guaranteed by the Constitution. It was more important that Congress's power to establish the post office was explicitly stated by law, because Congress was not able to do much at the time of the Convention.

tldr: Congress had extremely limited powers before the Constitution, and most powers were at the state level. It was important to list Congressional powers like establishing a Post Office explicitly in Article One, because things that weren't listed explicitly were left to the jurisdiction of the States/People

edit: words

The Post was the important long-distance communication in this period. So, when capehart_karsh sets out why the framers of the Constitution would see the need to specifically mention a Post Office in it, it was thought to be a very important thing to do. Note that the clause just says, "To establish Post Offices and post Roads". In the discussion of it, Benjamin Franklin clearly thought that the government would be doing quite a lot: he had previously been the postmaster of Philadelphia and Postmaster General of the Continental Congress, and he wanted the clause to even include building canals- the cutting-edge transportation technology of the day. This was rejected, as some delegates noted that the cost of building such a canal would be borne by the whole US, but the benefits would only go to the states that had the canal. This would later become the Internal Improvements debate of the early 19th century, where poorer states of low population density and few roads ( mostly the South) wanted the US government to undertake building roads because they needed them, which was resisted by richer states of higher population density ( mostly the North) which already had some roads. Because of that dispute, for quite some time the government would interpret the clause as just designating Post roads and largely stay out of building roads of any kind- with the exception of the National Road ,as initially it usefully connected the Chesapeake Bay ( at Baltimore) with the Ohio River ( at Brownsville, PA) and ran more or less between the North and the South.

The framers also thought the post office would be a source of revenue for the government, which would be needed: the previous Continental Congress had been pretty cash-starved. Possibly good revenue, because this was also before the present system of pre-paid mail: a letter would be typically handed off at a coach stop, then handed from passenger coach to passenger coach until it came to rest at another coach stop ( typically an inn) where there would be a dedicated desk for mail. At every change of hands, from one carrier to another, the letter would be paid for, and when it came to rest at the final destination ( typically another inn) the addressee had to pay for it- and the cost could be considerable. Partially for this reason most letter writing at this time was apparently done for business. By establishing a post office, therefore, the framers would be thinking they were both funding the federal government and also encouraging commerce- not simply helping people to be writing home to their mothers.