Why does my brake housing keep breaking
Whatever housing stop contained the pictured housing end has a hole that's too big.
Most BMX brake levers have barrel adjusters. If this is true for yours, and as you indicate this occurred at the brake lever end, replace the barrel adjuster. You need to get the right size, and there are several possibilities. Real BMX shops tend to have all the sizes, but replacing barrel adjusters all the time is less of a thing elsewhere so not every bike shop will necessarily have what you need.
If the lever doesn't have a barrel adjuster, as in certain fancy minimalist levers, you might need to replace the brake lever.
If a housing stop braze-on is allowing this to happen, the expedient fix is going to depend on where on the frame it is, what type it is, and whether persuading it back into shape with pliers seems to be a viable option. If it's a normal split housing stop then just reaming it out completely and using it to contain an M5 or M6 barrel adjuster to act as a housing stop may be an option.