From what I listened to, Dan Carlin seems to be referring to the "No Transfer Doctrine" or "No Transfer Policy" articulated by President James Madison in 1811. Basically, that Florida (which then belonged to Spain) could not be transfered to other powers by sale or other forms of exchange without the US considering it a hostile act.

After the US annexed both West and East Florida the principle was later understood to apply to other territories, such as Cuba, which were strategically valuable to the US. It is also often considered the key precedent for the Monroe Doctrine, articulated by President James Monroe in 1823.

The US, to my knowledge, never actually forced Spain to sign a legally binding contract preventing Spain from selling the island. Part of the reason for this is that Spain wasn't even planning on selling Cuba. It was economically important and, as one of the last key colonial possessions Spain had in the Americas, it was a matter of pride that it remain Spanish. They fought two long and bloody independence wars to keep it. It wasn't just about the money; for the Spanish it was a matter of principle.

When Spain lost its colonies after the Spanish American War the so-called Generation of 1898 inaugurated a period of national soul searching, economic recession, and national trauma. Spain's colonies had become a part of its national identity. Its dominion of overseas provinces part of its purpose. Now that most were gone, what was left? What unified the country? This is one of the themes that Jose Ortega y Gasset would later touch on in España Invertebrada (Invertebrate Spain) (1921), which attempted to explain the lack of national unity of the kingdom's numerous European territories and guess at what the country's national purpose should be.

There is a great deal written on this topic and period.

A good place to start is Louis Perez Jr.'s Cuba and the United States: Ties of Singular Intimacy, which is a slim volume that serves as a massive tl;dr for all the 500-1,000 page books on the subject.

A good companion to this is Raymond Carr's Modern Spain: 1808-1975. The book already assumes you know the basic gist of what's happening, so I'm not suggesting you read the whole thing. However, you can find the pertinent sections on the loss of Cuba and its economic and psychological significance to the Spanish people by looking up Cuba in the book's index.

As a final note, Dan Carlin is not a historian. I've heard many of my colleagues make complaints about the accuracy of his podcasts in the past when he focused on Ancient History but since I'm no expert on that topic I was mostly going on their word. Now that he is touching on topics I have some expertise in... I definitely get why so many historians criticize Dan Carlin's podcast. Think of it as entertainment that is useful if it inspires you to do more research on your own. Like a 'Based on Real Events' historical drama on TV. Take his statements with a grain of salt though.

Hope it helps.