How do plants store energy by potassium battery?
I'll give it a layman's attempt in sort of an "ELI5" approach. I hope the analogy isn't too simplistic, and please don't take it too literally.

How do plants use K+?

Plants control K+ concentrations and do work within the plant by pumping K+ between different compartments. Because water tends to follow solute concentrations (osmosis), plants can do things like suck in water simply by moving potassium around.

(note: animal cells do something similar: this is how your nervous system, kidneys, digestive tract, and other cells/organs perform their functions, but using different ions or combinations of ions)

Analogy to some human activities

Moving all this K+ around takes a lot of energy. Creating a concentration gradient is a lot like pumping water uphill: you put some energy in, but you are storing that energy as potential energy for the water flowing back downhill, or the ions flowing back against their concentration gradient (or water flowing the other way). But normally, plants use this as a way to spend energy to do useful things, rather than a way to store it. It's like they are pumping the water up the hill and then floating useful ores or lumber on a boat as it flows back down the hill.

What is actually happening that Wikipedia describes as a plant "K+ battery"

The article cited on Wikipedia, though, Gajdanowicz et al 2011, shows that sometimes plants take advantage of the mobility of potassium ions to assist the H+ ATPase that is another form of energy storage. Plants use H+ (proton) concentration gradients similarly to how they use K+ concentration gradients, but they are used to power different processes in the plant. Sort of like two different types of batteries.

Return to the analogy, and why this is useful to the plant

Normally both H+ gradients and K+ gradients are powered by ATP, which is indirectly produced via photosynthesis: think of it like the plant's coal-fired power plant. Converting K+ concentration gradients to H+ concentration gradients would be like using the water you pumped up the hill to power a hydroelectric plant that is charging a car battery. The benefit for the plant is that, because K+ can be easily moved around, the plant is able to power those hydroelectric plants anywhere around the hill, and that can be advantageous if a particular part of the plant is low on other sources of energy.