HD Wallet Child Private Key Derivation Function
A private key is a number, a very big 256 bit number. You can do ordinary arithmetic on such numbers, for example addition modulo n, where n is the order of the curve (someone else will have to expand on that if needed). n is very close to but less than 2^256.

Here's a diagram of the child private key derivation:



The left 256 bits of the 512 bit hash is added modulo n to the parent private key. The sum, which is also a 256 bit number (because of modulo n), is the child private key. The sum of the parent private key and a seemingly random 256 bit number is a seemingly random 256 bit number, so that sum will function just fine as a private key.