Just like website hosting, game hosting is what it sounds. You use a third-party service that provides the hardware and software where your game can reside, using their resources instead of your own. Such service providers (like Lemehost.com) can afford to utilize top-grade equipment designed specifically for that type of work, versus the average home computer that was probably never built to handle massively resource-intensive games.
In other words, game servers, or hosts, take on more of the workload so your poor computer doesn’t have to. As Game Skinny puts it, game servers are simply “a dedicated computer that owns a server code and provides many players access to entering the one game simultaneously.