Light, which travels at 186,000 miles per second in a vacuum, marks the speed limit of the universe. To quote the late, great Douglas Adams, the only thing that travels faster than light is bad news.
It's all wrapped up with Einstein's theory of relativity. If something travelled faster than light it would be going back in time and that would tinker with cause and effect. It would be a mess, frankly, and you can see why any sensible universe would apply such a speed limit.
It takes energy to go faster, as any runner will tell you. As you approach the speed of light, the amount of energy needed approaches infinity and this is how the universe enforces the speed limit.
Light is pretty slow, though. 186,000 miles per second seems fast based on our Earthly experience but the universe is a big place. Some of our telescopes are picking up light from distant objects that has been travelling for billions of years. If it takes light billions of years to get somewhere, it's going to take us considerably longer.
This is all very inconvenient for science fiction writers. If Picard wants to clobber the Romulans, he first has to get to Romulus and that would take ages at subluminal speeds. Most of the TV programme would consist of travelling and it would take so long that it would be Picard's great-great-great (x many) grandson who actually made it to Romulus. This would make Star Trek as tedious as a French arthouse film and nobody would watch it.
So science fiction writers have a choice. They can either just discard the laws of physics and travel by magic or they can respect the laws of physics and find a way around the speed limit. And there are ways, in theory.
One common method science fiction writers utilise is the warping of space. If you contract the space in front of you, effectively pulling your target destination towards you, then hop across this shorter distance and then unfold space behind you, you've got somewhere faster than light without ever physically travelling faster than light yourself. The universe's police would not be able to prosecute you for exceeding the speed limit and your starship's tachograph would prove that.
You haven't moved through space faster than light, you've just reduced the distance you need to travel by warping space.
Yes, you're probably saying, but surely this is all just science fiction isn't it? Well, it may surprise you to learn that proper scientists have researched the idea of a warp drive and some feel it's theoretically possible.
The gap between theory and practicality is however huge. For example, one paper on the subject proposes using exotic particles to warp space. Such particles would have negative mass, which is a very odd concept. The biggest problem with using exotic particles, though, is that they don't exist. Whilst certain types of exotic particles are theoretically possible, nobody has ever found one.
So we're unlikely to see Elon Musk creating warp drives any time soon, and maybe never. It is important to explore the possibilities, though. Even if we manage to avoid destroying ourselves and start to look after the Earth, we'll have to move eventually. The Sun will die one day and we don't want to be around when that happens. There may be sooner problems than that anyway — it's just a matter of time before some extinction-sized astronomical object heads our way.
The nearest star system to us is Alpha Centauri and let's assume for one minute it has a habitable planet. With present technology it would take us 80,000 years to get there. Even if we find a way to travel a hundred times faster than we can today, we're still talking about an 800 year journey. And that's just to the nearest star system.
It sounds like an insurmountable task at the moment but things are only impossible until they're not and these things are worth investigation because our long-term survival depends on them.