Carbon fiber composites are made up of sheets of carbon fiber filaments. These are usually impregnated with resin. To make CF structures, manufacturers will cut up the sheets, then lay them up onto a mandrel or other structure. They will then bake them at high pressure and temperature to cure the resin. The image below is from a Youtube video showing how Mad Fiber wheels are made - that company is now defunct, this is just for illustrative purposes. The metal item is a mandrel, which is a bit like a mold except that you put sheets of carbon on to a mandrel, versus you would put things in a mold.
If you have a CF frame or component and it's been damaged such that you can see torn fibers, it needs repair or replacement. This is obvious but probably worth stating. The image below is from Appleman Bicycles, who is a carbon framebuilder and repairer in my city. Disclosure, he's local to me and we are acquaintances, but I'm not a customer. On the left side of the frame, the fibers are clearly broken. It may not be as obvious on the right side, but I'm pretty sure that if you saw that frame in person, you'd see broken fibers also.
With that trivial question out of the way, it is frequently hard to know what to do if you crash on a carbon frame or component and there's no obvious damage. Damage to carbon items is frequently not visible at the surface. However, an impact, even a fairly soft one, might lead to a delamination below the surface that later grows into a larger failure. Delamination is the CF term of art for a fracture.
Ultrasound or other imaging is used to inspect CF parts in the aerospace industry. Ideally, we would have this capability widely available, but we do not. This is the only reliable way to detect a delamination under the surface. There may be a carbon repair shop close to you with this capability, but your local bike store cannot do this. Many people will propose a tap test, where you tap a coin lightly around the suspected fracture and listen for a change in pitch. The problem is that this test is most reliable on flat surfaces, and bikes don't really have a lot of those.
What to do: a proposed set of steps for consumers
This section concerns what to do if you sustain an impact that doesn't leave visible damage. For major impacts like a car vs. bike accident or a crash at racing speeds, more caution is recommended. It also depends on what part is involved; if your handlebars are damaged and they break, that will almost surely cause a crash, although if you are extremely skilled you may be able to recover (but you shouldn't count on being able to do this). I think the same would hold for stems and forks. I would lean towards replacing these parts immediately if you take a major impact. Minor impacts can be handled through watchful waiting, but some level of caution and regular inspection is still recommended.
I'm less certain about frames. In many cases, they are shielded from direct hits by the other parts of the bicycle and by your body. Again, the ideal case would involve an ultrasound or equivalent inspection after a crash. In the absence of that, I would propose watchful waiting. Be familiar with how your bike sounds and how it flexes under your pedaling. If you notice a sudden change, stop riding and try to see if there are any cracks. The picture below is from a video by Raoul Luescher, a carbon bike repairer based in Australia. Here, a metal part associated with the steering system hit the bike's head tube from inside due to arguably poor design. There was a small visible crack from the outside.
I believe that paint chips around frequently handled parts can be ignored. These might include areas around the dropouts. Gravel bikes may frequently take paint chips from flying bits of gravel. Some examples of what I believe to be paint chips are below, taken from recent posts.
Last, consumers should be aware that damaged carbon frames can be repaired. These repairs are not that expensive compared to repairing metal frames. On metal frames, damage will frequently involve replacing the entire affected tube. In contrast, with carbon, the second photo from Appleman bicycles shows that repairs can be localized to just the damaged area. I believe that forks, wheels, stems, and handlebars are not cost effective to repair; you should just replace those if damaged.
As a side note, carbon fiber does not have inherently poor potential lifespan. It has a very long fatigue life, although @NathanKnutson correctly notes that the engineering concept of fatigue in metals does not apply to carbon fiber composites.
A side note: minor surface damage inside the frame
As I discuss on another answer, frame manufacturers designing for high-level competition obviously push frame design. There may be cases when a carbon tube is flexing against a metal item. For example, in late 2021, Specialized recalled their Tarmac SL7 frames and forks. The headset had a compression ring on the outside of the fork steerer. As the bike hit bumps, the compression ring had an relatively sharp edge that dug into the steerer tube's surface. This would eventually cause a failure. However, Raoul Luescher raised an issue with the fix, namely that now the compression plug, which was cut to length, could have a sharp edge as well. He showed a borescope shot of the inside of the fork where the edge of a compression plug marked the surface of the fork, and he argued that this is likely to fail eventually. For performance road bikes with hydraulic hoses routed inside the steerer, an unexplored issue is that over time, the hoses might abrade through the steerer tube's fibers as well.
The watchful waiting advice applies to damage from one-time events. The issue of damage from repeated events is a more serious one. However, this sort of damage is likely to happen outside your line of sight, and it might require specialized tools to even see.