Physical Explanation for variation of Gibbs Free Energy with Temperature and Pressure Reading Atkins Elements of Physical Chemistry right now, and the book says:
"At constant pressure and entropy, an increase in temperature ($\Delta T\gt0$) results in a decrease in $G_\mathrm m$ (molar Gibbs energy) ($\Delta G_\mathrm m\lt0$)."
"At constant molar volume and temperature, an increase in pressure ($\Delta p &gt;0$) results in an increase in molar Gibbs energy ($\Delta G_\mathrm m\gt0$)."

They have equations to describe this but I'm not really interested in the mathematical derivations (those are understandable) but rather in a logical, physical, and qualitative explanation of why these relations occur.

Given that $\Delta G$ indicates the maximum non-expansion work derivable from a system, what does this mean for the above relations? Does increasing the pressure of the system require work input ($\Delta G$ is positive) while increasing temperature result in the system doing work/an output of work ($\Delta G$ is negative)? 

So, given a system (like a gas or something), why does $G$ change the way it does in response to $T$ and $p$ changes, and what does this physically mean for the system?

I'm a high school student (went through AP Chem) who's just trying to delve deeper into chemistry, so the simplest and most "layman's terms" explanation would probably help.
Probably the easiest way to picture it is that, since the Gibbs function is indeed a measure of the potential of the system to transfer energy in certain ways, these transformations affect the energy that can be transferred.

For a certain system at constant pressure and entropy (which therefore cannot absorb thermal energy from its surroundings), an increase in temperature is necessarily linked to an internal transformation - an internal work - that reduces the potential energy available to exert work on its surroundings; that increase in temperature is "using up" part of the potential reversible work as measured by the Gibbs function.

On the other hand, a system at constant volume and temperature which increases its pressure (and which, given these constrains, is therefore receiving some form of energy input from the outside; note that there isn't a pressure-volume tradeoff) will increase the potential reversible work it can exert on its surroundings, as measured by the Gibbs function.

It is important to note the implications of the constant conditions that each of those two cases set, because these constant conditions are telling you which forms of energy transfer are permitted or prohibited in each case.