Many people in the personal hydroponic and AeroGarden community refer to these passive hydroponic jars as "Kratky jars". They are loosely based on the Kratky method of passive hydroponics, but technically they aren't. The actual Kratky method is a 'fill and forget it' method – that is, once you set everything up, you shouldn't fill the water reservoir again. This is a key component in the method, while these jars are much too small to actually do this. That said, this vaguely Kratky-esque method is very popular and effective.
In the AeroGarden community, they are generally used to move plants grown in the AeroGarden out into the jars, generally the last stragglers of a crop, to enable you to replant the garden with a new crop, while taking advantage of the AeroGarden's "spillover" light. However, sometimes you want to start plants in addition to the ones in your garden, and so you need to start them direct from seed. Here, I will lay out the process I used to start some herbs I wanted to grow.