Q: "What exactly are “repositioning cruises”?"
A Reposition Cruise is the sailing taken by a vessel when changing homeports. While they are sometimes marketed as special guest experiences, a repo cruise is an operational task.
The line will reposition as ship to follow demand, often to a relatively nearby port, but the repositioning cruise is a different thing.
Important point, a Repositioning Cruise is an operational task for the line, it is not a consumer offering. So, with quite few exceptions, RCI for instance, you will not find cruises marketed as "Repositioning Cruises". This is because, well, I guess "Repositioning" doesn't sound very interesting or fun or vacation friendly. Instead, they are marketed as specific types of voyages such as Transatlantic or Panama Canal. Or, most common, not marketed at all. For example, a repo cruise from Baltimore to Galveston will be marketed as a Caribbean Cruise, not a Repositioning Cruise.
Q: "It's something to do with cruise ships being moved across the ocean just before the cruise season starts and just after it ends and they offer super low prices since it's out of season."
NO. The distance involved is irrelevant. For example, ships are repositioned on the East Coast of the United States routinely.
Also, transoceanic cruises are often sold at a premium for destination ships such as Oasis (RCI) or Breakaway (NCL) Class ships. Small ships may appear discounted though.
Q: "Is this something done worldwide by all cruise lines or just by one or two in a certain area?"
YES, for major cruise lines. Regional lines, less so due to narrower scope.
Q: What are the very basic facts that all budget travellers should know about this seemingly almost secret means of travel?
First, there's really nothing secret about them. They are very common. Royal Caribbean markets Reposition cruises specifically for significant itineraries. But, other repo cruises are marketed as specific types such as Transatlantic. But again, many repo cruises at not noted at all unless you notice the difference in embarkation and debarkation ports.
But even that's not always an indicator, for example, Norwegian Epic will be sailing a Med circle where they will embark at both Barcelona and Civitavecchia but her home port technically Barcelona so her upcoming repo will be from Port Canaveral to Barcelona.
Just because a cruise is a repo does not mean it will be a 'budget' option. The line will sell passage at the highest fare they can. For example, East/West Coast US repos will command a much higher fare due to the Panama Canal transit. Specialty not considered, older, mid-sized ships would offer the best value.
Also note that while the line will sell passage on different segments, a round the world cruise rarely repositions the ship. For example, Queen Elizabeth's next RTW cruise departs and returns to Southampton despite offering ~15 one way Itineraries along the way.