Most FTL theories that don't involve hand-waving require an enormous amount of energy and an enormous amount of mathematical calculations to avoid obstacles. For example, imagine travelling at 5 c and you run in to a star you couldn't see coming. Game over. For you, the star you ran into, and its surrounding solar system.
Authors usually gloss over the details, but it's implied that the energy required is more than say, the world's annual production of electricity (or perhaps even more than humanity has generated in all history), or more CPU calculations than the entire world could calculate in a lifetime on modern hardware. Some obscenely large value that they anticipate we could theoretically reach one day.
The massive energy requirement usually comes from needing to "fold" or "tear" space-time, usually based on scientific theories by physicists or other authors. In other words, the ship itself does not travel at/greater than the speed of light, but it covers distances that qualify as FTL from a single observer's point of view. This is usually because, during research, authors discover that the mass required to actually travel faster than the speed of light would be beyond infinite, so it would require far less energy to alter space instead of trying to move through it.
Sometime it's a wormhole, other times it's accessing "sub-space" or "hyper-space", and yet others a "warp bubble" (a bubble of gravity waves, which NASA is actively researching...), and rarely is some more exotic form of travel, like a tesseract. No matter what it is, though, it is implied that it is some technological feat beyond what we can do today, from generating copious amounts of anti-matter, to charging up something like a zettawatt of electricity and releasing it all at once. The more impressive it sounds, the better. The point is, unlike pure fiction (say, a crime drama with real physics), we only have conjecture, so authors try to make it plausible, even if it's really only theoretical.
Finally, it should be noted: drama. If you can just push a button and instantly end up have a universe away, there's no tension, no drama (or, it's just not part of the story, so they hand-waved it). Instead, you need 10 seconds to calculate the necessary route, but you'll be caught in a supernova in 9 seconds. Or, you need to dodge enemy fire/fighters while the computer calmly tells you that it's working as fast as it can, several shots narrowly missing a nacelle. You get the idea. You'll notice how FTL seems to take no time at all when they're not in combat or racing to save someone's life, but takes forever otherwise.