The main thing about Magna Carta (was about to write
the Magna Carta, old habits die hard) is that it was a restriction on what, at the time, was the unfettered sovereignty of the king. It was people -- admittedly, rich and powerful, land-owning people -- putting a small brake on a very powerful force. From this the philosophy of a government being accountable to the people, or at least not 100% immune from scrutiny, eventually arises.
The framers of the US Constitution had classical educations (anyone educated, at the time, did) and so they were familiar with not only history but also philosophy, so they had read up on natural rights, hence the idea of original, inalienable rights, begins to arise, too. But make no mistake, the framers were also wealthy landowners, and most of them wanted those fine inalienable rights to mainly go to wealthy landowners. There was plenty of debate about slavery when creating the Constitution (see:
Founding Brothers: The Revolutionary Generation by Joseph J. Ellis for some more about the slave debate during Revolutionary times --- but even the folks who were opposed to slavery were more interested in a gradual withdrawal than anything else, because they knew that that tactic was the only possible way to attract the mid-Atlantic states and possibly get more support for that position (the gradual withdrawal of slavery really didn't fly; states that wanted it gone outlawed it whenever their legislature got around to outlawing it. Those that didn't eventually seceded in the 19th century). Yep, playing politics even then -- horse trading to get your bill passed is nothing new.
But getting back to the basis of American law: later law is based on earlier law, subject to interpretations which cover a lot of areas of scholarship, such as philosophy but also pragmatic considerations, but it all heads back to the Constitution because that's the final arbiter: whether something is Constitutional. And the Constitution itself is based upon philosophy and older British law. The framers (not exactly the same people as the ones we call the founders) were well aware that it would be a living document and would change over time, and they took that into consideration, but a lot of them had their pet concerns (hey, just like modern politicians!) or preferences and that creeps into the discussions. The Federalist Papers are not the only source of their thought processes -- there are letters, journal entries, etc. The origins and original intent behind the Constitution are no simple thing to unravel.