What causes egg-covered puffed rice cakes to become soft when pan fried (and how to control it)?
They get softened up when the moisture from the egg gets hot enough to bathe the rice cake with steam, and rehydrate the starches with moisture and heat.

The rice cake was absorbing moisture while it was sitting in the egg - not too much if it still seemed light, and clearly not enough to start disintegrating, but some.  The texture of such dried products is open, but the strands (of gluten, or starch) are dried hard - the crumb can be heavy with moisture if soaked long enough, but it drives out the air not soaks into or softens the starch.  They will still be stiff (and dripping) after soaking, and they won't ever become soft and chewy from all the way dried out, with only added moisture.  

Add heat, though, and the combination lets the inner structures absorb the moisture from the surface, reconstituting the starches to a chewy consistency (the brittle stiff structures swelling, and becoming flexible and compressible like bread and not just brittle or mush like soaked crackers) - and it becomes flexible enough to let the steam wick upward through the rice cake to bring the combination of moisture and heat to transform the interior.  If the moisture form the soak hadn't reached the center, though, it can remain  dry and stiff - especially if the edges also hadn't soaked enough to get water for the steam, which helps mobility (turns it from soggy-edge and dry-center to moist all through).

This process isn't limited to eggs, or to rice cake.  If you have bread that's gone bone dry and hard, soaking will soften the edges to mush but it won't reconstitute it to a bread-like consistency without heat.  Soak it in water or milk, though, then pan fry or toast it - and the heat and moisture will make the gluten absorb the water, swell, and become flexible again.  I have rehydrated bread, rolls, or pastries this way (usually with water, it changes less) if they become totally dried out without otherwise spoiling - soaked in water for a while, then heated to transform from wet-and-brittle to soft and chewy.

You should see similar effects in your rice cakes (becoming soft/chewy) if you soaked them in water or milk or whatever, before frying again, to those you got when dipping them in egg first, and again similar effects from using a toasting or using an oven rather than frying.  The combination of heat and moisture makes a difference in re-hydrating starches, not just the specific combo of egg dip and ricecake (disclaimer - obviously, there will be different results from using egg, water, milk, or alcohol, and rehydrating rice-cakes, breads, and other dried starches - just, similar kinds of results from moisture, heat, and starches)