How do Bitcoin mining pools send their miner's their share of the block reward In most bitcoin mining pools, miners of a pool are sent their portion of a block reward that the pool collectively mines, every 30 minutes or 1 hour. 

How exactly does a mining pool tell the Bitcoin network to send these coins to each of the miner's addresses?
They do so just like anyone else sending Bitcoin to someone; they create a Bitcoin transaction and broadcasts it to the network. It works in exactly the same way as you sending Bitcoin to someone else. The mining pool operator goes into the pool's wallet, chooses the "send" option in that wallet, and enters the addresses and amounts to be paid for each miner. Of course this is all automated so the mining pool operator doesn't actually do that. Instead a script does it and interacts with the wallet using RPC calls.

Sometimes the mining pool may choose to pay a low transaction fee or not broadcast the transaction and instead include it in the next block the mining pool is working on.