What happens in people with obstructive sleep apnea?

When we sleep the throat muscles relax and become floppy (like other muscles). In most people, this does not affect breathing. If you have OSA, the throat muscles become so relaxed and floppy during sleep that they cause a narrowing or even a complete blockage of the airway.
When the airway is narrowed and the airflow is restricted, at first this causes snoring. If there is a complete blockage then the breathing actually stops (apnea). The blood oxygen level then goes down which is detected by the brain. Your brain then tells you to wake up and you make an extra effort to breathe, and you start to breathe again with a few deep breaths. You will then normally quickly go back off to sleep again and not even be aware that you have woken up.
If someone watches you, he or she will notice that you stop breathing for a short time, and then make a loud snore and a snort, perhaps even sound as if you are briefly choking, briefly wake up, and then get straight back off to sleep.
It is quite common for many of us to have the odd episode of apnea when we are asleep, often finishing with a 'snort'. This is of no concern. In fact, some people when they sleep have periods of 10-20 seconds when they do not breathe. However, people with OSA have many such episodes during the night. In fact, the episodes of apnea are usually only considered 'significant' if they occur more than 10 times per hour, and for each spell of apnea (stopped breathing) to last 10 seconds or more. People with OSA can have hundreds of episodes of apnea each night.
So, if you have OSA, you wake up many times during the night. You will not remember most of the times but your sleep will have been greatly disturbed. As a consequence, you will usually feel sleepy during the day. Daytime sleepiness in someone who is a loud snorer at night is the classic hallmark of someone who has OSA.