How does codeine affect our brain's neurotransmitters?

Slows. Narcotics typically slow neurotransmission, especially the ones signalling pain.
Similar to endorphin. The chemical structure of Codeine is similar to that of the neurotransmitter, endorphin, a natural pain-killer produced by the body. As such, Codeine molecules can fit readily into endorphin receptor sites, and stimulate those neurons to accomplish the same end result - suppressing the perception of pain.