To add to what you said engine braking allows the engine to intake air as normal but then it keeps the exhaust valve closed as the piston starts moving up which compresses all the air and then at top dead center it opens the exhaust valve. Letting out all that compressed air at once is why it makes the loud noise.

There's typically 4 stages for a 4 stroke engine that it goes through. Intake - Air is sucked in from the intake. Compression - That air is compressed and at some point fuel is injected. Combustion - the air fuel mix combusts creating power. Exhaust - the piston pushes out the exhaust gasses. Repeat.

The reason the compression stroke doesn't normally waste tons of energy is because you don't open the exhaust valve at the top of it, so you get back most of the energy you put in as the piston goes back down during the combustion stroke.

Also when engine braking is on there is no combustion or exhaust stroke. It just does compression with exhaust at top dead center followed by intake and repeated that cycle.

There's also exhaust braking which is slightly different and normally only used on smaller trucks. It's quieter but provides much less braking force. Basically you just restrict the exhaust so there's high back pressure all the time, air still gets through so you don't reclaim the energy but it doesn't build nearly as much pressure so it doesn't brake nearly as much but it's also not nearly as loud.