Here's the general idea:
The reptilian brain has all the components of "running a smooth engine". It regulates breathing and heart rate based on chemoreception of CO2 and O2 as well as basic motor motions, olfactory, and gustatory that allow the organisms to find, chase/kill, and eat appropriate foods. It can also deposit hormones into the body via the pituitary gland (i.e. adrenaline response).
Limbic system is often referred to as "early mammalian" system. And neocortex is more of a later mammalian thing, becoming a giant part of the brain in species like hominids (including humans), dolphins, and elephants.