Inks, paints, and dyes consist of particles of colour that are dissolved or suspended in a liquid base. When you write, the liquid part dries and leaves just the colour behind.
In this activity, when the water creeps up the filter paper, it contacts the dried colour. The colour molecules in the ink are dissolved and carried up the strip. The more soluble a molecule is in water, the higher it will get carried up the paper. Different colours get carried further and faster than others. The permanent marker and dry-erase board inks are not water soluble, so the water will not be able to dissolve its pigments to carry them up the strip.
In this mystery, the culprit has left behind a ransom note and the students are asked to determine which of the 4 suspects wrote the note. Each suspect owns a different marker. Each brand of marker will have a different combination of pigments; by separating out these pigments and comparing them we can find out which pen was used to write the note.