finding them
Butterflies drift around Forests during daylight and vanish with the sun. If none are out, shaking Forest or Pearlwood trees while it is still day gives you a chance at one. How many the world produces gets rerolled at dusk and again every time the world loads, so some days the treeline is thick with them and other days there is nothing at all. Celebration Mk 10 worlds run heavier than normal.
why they are worth stopping for
For a fresh character, a good butterfly day is real money. Catch them with a Bug Net, sell what you do not want to keep, and a single afternoon can outearn anything else available at that stage. The ones you hold onto go into jars for decoration or straight into the bait slot, since the whole family works for fishing.
as bait
Bait quality varies a lot between species, and the Monarch is the worst bait in the entire game, a nod to the real insect tasting foul to anything that tries to eat it. Rarer colors are worth far more both in the tackle box and at the shop, so check what you actually netted before you dump the stack on a vendor. A Gold Butterfly is one to keep rather than burn on a fishing trip.
the hell butterfly
There is an Underworld version, out during the day like its surface relatives, that also pops loose when you cut down wild ash grass. A standard net will not take it. Swinging one at it burns your hand and sets you on fire instead, so bring the Lavaproof Bug Net or the Golden Bug Net before you go looking. Only a small number of Hell Butterflies and Lavaflies can exist near you at once, so if nothing new is appearing, clear out the ones already flying before you wait around.