How does sleep apnea interfere with your sleep?

Prevents deep sleep. Apnea causes an arousal in the brain that either 1) wakes you up or 2) lightens the stage of sleep you are in. Sleep is like walking down the stairs... It's a stepwise approach to deeper sleep. The more severe your apnea, the more you "go back up the stairs" to lighter sleep... Continued interruption may prevent you from "getting all the way down the stairs.".
Prevents deep sleep. A person with sleep apnea will partly wake up many times during an hour which prevents that person from getting deep sleep. Deep sleep is necessary for the body and mind the restore itself. Without it we will feel tired and our health is compromised in multiple ways.
Many ways. Sleep apnea is when you stop breathing during sleep. This prevents you from getting into the deeper stages of sleep where you get the beneficial sleep during which the body rejuvenates. It can lead to insomnia. Your blood oxygen level goes down and your heart beats faster. When you stop breathing, you may wake up with a gasp struggling for air. You may not even remember this occuring.
Mechanism. Sleep apnea is an involuntary cessation of breathing that occurs while the patient is asleep. There are three types of sleep apnea: obstructive, central, and mixed. Of the three, obstructive sleep apnea, often called OSA for short, is the most common. The obstruction decreases oxygen in your blood and arouses you from sleep.