I am a petroleum engineer, I will try to help with a VERY BASIC explanation of what a PETE does.
There are 3 basic sub-disciplines to petroleum engineering. Drilling, production engineering, reservoir engineering.
In undergrad you learn about all of these to a certain extent but TAMU leans toward reservoir engineering a bit.
Drilling is just that, how to drill. Horizontally, vertically, directionally, through different types of rocks, pressure regimes, etc.
Production engineering takes over when the well hits bottom. These guys design the completion, any stimulation that needs to be done to the well, and ensure that the well is brought on line. They also monitor the wells' performance and intervene if there is a problem.
Reservoir engineers manage the reservoir from a larger perspective, which may have one well or a thousand wells in it. They study fluid flow through the rock, look at how to better manage production from the reservoir as a whole, etc.
Now, there are many other sub-disciplines in these categories. And there is an awful lot of paper-chasing just like in any other job.
The salaries are high because the average age of the workforce is over 50, and in a few years there will be a dire shortage of engineers. Also, companies are sniping talent from one another, so they are paying more and more to retain their people rather than watch them go to the highest bidder.