That’s a great question! Scientists are still trying to work out exactly what makes bears enter and leave hibernation. There’s probably a combination of external factors (like the temperature of the environment) and internal factors (like hormone or nutrition levels).

First, it’s important to understand what hibernation is about. It’s an adaptation to living in a climate that has big seasonal variation. Usually, it means a cold winter with low food availability. Hibernation is a sort of super-sleep. A hibernating animal slows down all of its processes (its metabolism). The slower the metabolism, the less energy they burn. Heating ourselves (as birds and mammals do) takes a LOT of energy, so hibernators drop their body temperature. Some animals actually have body temperatures below freezing in winter, but bears don’t drop their temperature as much. If the timing is right, they enter hibernation after they have put on a lot of extra fat to fuel themselves during the winter, and before food gets scarce. They should leave when there’s enough food around to support them.

When you think about it, external factors would be important for helping bears survive in different environments. Let’s just look at black bears. They can be found from Mexico to Alaska. Obviously, the conditions are very different in the northern and southern parts of their range. If their hibernation clock were completely determined by genetics, southern bears would hibernate too much and northern bears would go into hibernation too late and leave too early. That means that they must be born with a flexible “hibernation clock” that can be adjusted by things like temperature of the environment. There seems to be more to it than just temperature, though. Availability of berries explains some of the differences in when bears emerge.

Even in the same location, bears may enter or leave hibernation at different times. For example, female bears give birth during the hibernation season. The female keeps hibernating, but the newborn cubs just snuggle up, suckle, sleep, and grow. If they hibernated, their metabolism would be low and they couldn’t do the work of growing. Females with new cubs stay in hibernation the longest. This probably allows the cubs to be bigger. Adult male bears come out of hibernation first. They may need to do this to get an advantage over other males. A bigger, stronger bear will be able to defend a bigger territory. Then females with no cubs or cubs from earlier years emerge after the males.

There’s some great information at this site:
nature. Did you know that bears may move a ton of dirt while digging their dens?

What else do you think is important in causing bears to enter or leave hibernation? You may want to study physiology if this kind of question interests you.

The mechanism is still not completely understood, but bears have something called a circannual clock. Just like we get sleepy when nighttime comes around, bears get sleepy and ready for hibernation when it starts to get cold. They know to wake up when it gets warm out. Technically, the hibernation period relies partially on the weather. This means that if there’s a really long summer, or spring comes early, their hibernation period could be very short!

Every animal has an internal “clock” in their brain that tells it when to fall asleep and wake up each day by using our senses. We have this clock and it tells us when to go to bed every night and when to wake up in the morning - our clock revolves around the sun. For animals that hibernate, like bears, they also have a “hibernation clock” in their brain that tells them when it’s time to hibernate and when it’s time to wake up. Bear’s brains have a stopwatch that sends chemicals to their brains when it is time to get up!

There are several good answers to this question here. In short, a hibernating animal might be able to detect seasonal changes in temperature, light, etc., or chemical changes such as those related to shortage of stored fat reserves, any of which signal that it is time to wake up. Some animals have genetic "alarm clocks" which set the start and end of hibernation, regardless of available food supplies (bears are one example of this. In this site go to ‘Bear Facts”).

On the other hand, animals may hibernate for reasons other than seasonal changes. Food shortage due to natural events (e.g. wildfires and storms ) can trigger reductions in metabolic activity to conserve fuel, and some avoid excessive heat.

Some additional information that might be interesting: a comparison (Link about “Bear Facts” above) of bear and chipmunk hibernation, and general information about hibernation.