"A micelle is stable formation of amphipathic molecules, or molecules that have a polar head and a nonpolar tail." Huh? Even that is more information than you probably need about micelles (even if you do understand it).
Micellar technology uses surfactants (fancy word for cleansing agents) that have both water-loving and oil-loving properties called micelles. These two parts help the molecules in your shampoo or facial cleansing products to attach to oil and dirt particles, which makes it easier to wash away. In short, the micelles are helping to get your hair or skin as clean as possible without being too aggressive. Think of them as a magnet: they help draw out impurities like dirt, oil and environmental pollutants without being too harsh.