I watched a documentary about how Christopher Columbus 'discovered' Cuba and there were Cuban natives there, and it got me wondering how on earth they got there. Could anyone help?
The (very) short answer to the question is that the natives of the Americas migrated from Asia to the Americas via the Bering Strait, during the Last Glacial Maximum (about 16,000-13,000 years ago). During that time, sea levels were substantially lower and there would have been dry land between present-day Russia and Alaska. The theory then is that as glaciers receded, there were ice-free routes from present-day Alaska into what's now Canada, then further south into the Americas.
Now that said: while this theory is agreed upon in general, the specific details of how this migration happened are highly contentious -- among other things, there are debates over how much migration happened on foot vs. by coastal routes, how many waves of migration there were, how quickly the Americas were settled, etc.
Here are some older posts on the subject to get you started:
https://www.reddit.com/r/AskHistorians/comments/49dje1/earliest_peoples_in_north_america/