Keto is a fairly simple concept to understand:
Research studies generally peg under 50 grams as ketogenic, and 50-150 as "low-carbohydrate". When most people eat less than around 50 grams a day of carbohydrate, they soon start producing high levels of something called "ketone bodies". These are simply breakdown products of fat, and are used to fuel cells that normally depend on glucose.
Keto's impact on health isn't so simple.
Your brain and nervous system usually run on glucose. Fueling it mostly with ketones is a whole new world to these cells, and may provide benefit. The main condition this has been studied for is epilepsy, but more research on other conditions is underway.
Keto's impact OUTSIDE of that, though, is more complicated. Keto is purported to aid with weight loss more so than other weight loss diets. The evidence for this is mixed, but tends to show fairly similar weight loss as other diets, according a meta-analysis of 32 trials comparing carb levels. Keto benefits for a variety of conditions may stem from restriction of carbohydrate, avoidance of typical carb-containing processed junk foods (so not the carb itself, but the food that contains it), and/or benefits from the ketone bodies themselves.
There are three different purported reasons that keto might not be totally safe:
First is the claim that high levels of ketones can be dangerous. This is generally not true with ketogenic diets, as ketoacidosis (dangerously high ketone levels typically caused by diabetes) doesn't seem to occur absent an uncontrolled medical condition, with no cases being reported in a two-year ketogenic diet study. However, possibly the first case of ketoacidosis in a nondiabetic patient was reported in 2015, in a lactating woman on a ketogenic diet.. A later case study purported to be the first of a healthy, nondiabetic, and non-lactating woman experiencing ketoacidosis.
Ketone bodies are acidic by their very nature, although blood pH is typically well-regulated by our kidneys and lungs. While there may be a theoretical detriment to mild acidosis, trials haven't shown adverse effects of sustained production of ketone bodies over time.
Second is the idea that going too long without getting "enough" carbohydrate is inherently dangerous. This is also generally not true, as carbohydrate isn't an essential nutrient (you can make all you need to survive, by a process called gluconeogenesis, without ingesting any through food or drink). There isn't any trial evidence though on what the long term effects (e.g. 5-10 or more years) of carbohydrate deprivation are, and whether ketogenic diets have any negative impact on unstudied health conditions, certain genotypes, etc.
Finally, there are claims of harm from diets high in animal products, and most keto diets happen to be high in animal products. This claim is nearly impossible to evaluate with a simple yay or nay, given the huge number of studies involved with varying conclusions. What we know is that evidence is mixed, and slightly differs depending on the animal product in question (for example, fish consumption often shows a slight benefit for longevity in observational studies). Multi-decade randomized trials aren't feasible for foods, so we have to rely on imperfect evidence. In prospective (non-trial) studies, processed red meat often has a strong correlation with increased all-cause mortality and diseases, whereas unprocessed red meat often has a much smaller or no correlation. Egg consumption typically shows negligible or no detriment to all-cause mortality and disease,  although again, evidence is mixed, with some other reviews finding a correlation of high egg intakes to disease.
Meta-analytic results can differ depending on timeframe (intermediate versus longer term outcomes), which studies are excluded and included, and a variety of other factors. Some studies are funded by financially-interested parties (for example, the beef and egg industries), raising the possibility of publication bias or other biases. But other non-industry funded studies also tend to back up a detriment primarily from processed meats, and less or negligible detriment from unprocessed meats, depending on the amount eaten. This will remain an open question, given the difficulty in conducting long term randomized diet trials.