To start off the chain of events, travel to Skyloft and speak with Parrow in the plaza near the Light Tower. He’ll mention that Strich found an island inside the Thunderhead that could serve as a bug paradise.
Fly to the Thunderhead and go inside. Head towards the southwestern portion, to the large island with the crumbling tower: Bug Rock. Speak with Strich there to learn about the Bug Heaven mini-game (you can play it if you want, but that’s not part of this).
Return to Skyloft and enter Beedle’s Airshop. Sleep in his bed until night to awaken on Beedle’s Island. Speak with Beedle outside to learn that his rare Horned Colossus Beetle is missing. Tell him you’ll find it for him. Go back into the shop and sleep until morning.
Upon awakening, leave the shop and fly back to Bug Rock. Strich there has Beedle’s beetle, but will not return it. Unless you can catch ten bugs in three minutes. (Now you know why you want the big Bug Net!)
Once you succeed, you’ll receive Beedle’s Beetle Cage. Return to Skyloft and head into Beedle’s Airshop. Again, sleep inside the shop until night to be transported to Beedle’s Island. Speak with Beedle outside for your nice reward. (Well, considering how some of his wares cost over 1,000 Rupees…)