There are lots of bugs everywhere. Collecting them is encouraged, but I don't see a use in them. They aren't needed in crafting or sellable to the peddler at night like materials, it seems. I'm obviously still gonna collect them all, but why am I doing it?
You could use the bugs you find to infuse the potions you buy to become more useful in your adventures or sell excess bugs to Strich in his room at the Academy at night.
To infuse your potions, you first need a bottle full of a potion purchased from Luv, the Potion Shop lady at the Bazaar. Next, take your potion to Bertie, her husband that's holding the baby to get your potions infused. He will tell you what bugs are needed to infuse your potions and how much rupees it will cost.
You could also sell them to Strich, the tall bug guy who is/was one of Groose's lackeys. If you visit his room in the academy at night (the one closest to the lower entrance), he will offer to buy some bugs from you. He will only offer to buy three types of bugs at any one night. So if he's not interested in buying ones you want to sell, come back another night and hopefully he will be interested then.
You can upgrade your potions with bugs, which is very helpful.
For instance, the red potion normally only refills 8 hearts. If you upgrade once by giving the potion lady's husband (the guy with the baby by the potion shop) some bugs, it instead refills all your hearts. If you upgrade it again, it not only refills all your hearts, but now you have a double portion in one bottle, so it can be used twice. Good deal!
Similarly, the shield repair potion repairs your shield and refills 4 hearts. But once you upgrade it, it'll repair your shield and refill 8 hearts, and you can double upgrade that one, too. Great for bringing into a temple.
Alternatively, go to Strich's room at night in the knight academy, and he'll pay you a pretty penny for some of those bugs. I believe he eventually creates an island later in the game that has tons of bugs for you to easily catch.
Some bugs sell for 40 rupees, like the easy-to-catch Eldin Rollers (Earth Temple), and Faron Grasshoppers (Lake Floria waterfall). Also the Volcanic Ladybugs are good for cash since there's some on the wall right outside the exit to Volcano Summit.
If the ones you want to sell aren't listed at Strich's, sleep in his bed until night and he will want different ones usually.
Also, the Bug Rock / Bug Heaven minigame (in the Thundercloud) will NOT let you keep the bugs you catch, instead when you get a decent time you are awarded with 5 random bugs - depending on what difficulty level you picked. Fastest way to get many harder to catch bugs.