Imagine a box, with a partition in the middle, separating warm and cold air. Take the partition away, let them mix, what happens? The denser, cold air rests on the bottom of the lighter, warm air. There was baroclinic instability in the original configuration, which was released when they sorted themselves out.
In the real world (Northern Hemisphere), the cold air is the north and warm is south. If a meridional wind were to flow through an area where there's a relatively sharp temperature gradient, this would 'kink' the isotherms (1000-500 thickness lines) and, in a sense, 'remove the partition' between the cold and warm air, like in the box. This results in a stronger east-west temperature gradient, and usually kicks off cyclogenesis (could be from lee-troughing, upper-level vorticity advection, etc.).
This E-W temperature gradient is what drives mid-latitude synoptic-scale storm systems. That's because warm air on the East promotes ascent and storm development through warm air advection, and vice-versa in the West.
Basically, baroclinic instability is all about temperature gradients. The kinetic energy that drives baroclinic activity comes from meridional wind flows, and the potential energy is all in the East-West temp. gradient. This release of baroclinic energy can start cyclogenesis (through kinking) and promote further development of the system.
Just for fun.. 'bar-' refers to pressure while 'clinic' refers to fixing or healing. The atmosphere is unstable around strong pressure gradients.. formation of storm systems is one way the atmosphere can get rid of excess energy. Just like someone might pace around when they're nervous. Make it make sense to YOU.
I'm open to constructive criticism or correction if anyone reads this btw!