tl;dr: Gitmo is located on a US naval base which pre-dates the trade embargo by half a century. The site was chosen for the detention centre chiefly because of the complex legal status of the base.

Guantánamo Bay (the geographical feature) has been a strategically important site for trade and warfare since the Spanish arrived in to colonise Cuba in the fifteenth century.

During the Spanish-American War of 1898, the US intervened on behalf of Cuban anti-colonialists who were fighting for independence against Spain. US Marines landed at the bay and won a crucial battle there.

The Americans didn't want to just let Cuba go, and so they occupied the island militarily for three years. They only agreed to withdraw their troops after essentially rewriting the new Cuban republic's constitution.

Known as the Platt Amendment, this rewriting allowed the US to militarily intervene in Cuba as it pleased, to control its economy and its diplomatic relations, and to set up naval bases and coaling stations (where you can refuel ships) on the island.

That last one is the most relevant here, because Guantánamo Bay Naval Base started life as one such coaling station in 1903.

In 1934, the Platt Amendment was repealed and replaced with a new treaty, which gave the US the right to lease Guantánamo Bay Naval Base in perpetuity for $4,085 a year.

Incidentally, the US still sends a cheque to the Cuban government for this amount every April, but they've not cashed it since the revolution in 1959. Fidel Castro and his successors all view the US presence at Guantánamo Bay as an illegitimate occupation of their territory.

So, why put a detention centre there? It comes down to the complicated legal status of the naval base. The US argument is that Cuba is formally sovereign over the base, and so US law, the US Constitution, and US obligations to international treaties don't apply.

The other part of the US argument is that, even though Cuba has formal sovereignty over the territory, Cuban laws don't apply there either.

Essentially, this allows the US military to act with impunity within the base. Academics have referred to this as a "legal black hole".

If you think the US government's argument is a complete logical fallacy, you're right. But there's nothing you or anyone else can do about it - even the President of the United States himself. This is what power looks like. This is what empire looks like.

As you can see, the US presence at Guantánamo Bay pre-dates the trade embargo of 1962 by a long way. This is why the location was chosen, and why the embargo had no effect on the detention centre.