From Nookipedia, the Animal Crossing wiki
Mole Crickets (オケラ, Okera) can be found during the winter and spring, making a low, repetitive noise underground, from which they can be identified. Players will have to dig in the area in which the noise is loudest. Having been uncovered, the Mole Cricket scuttles away and if left alone for long enough, it escapes by burying. It can drop into a body of water as any crawling insect. They are sold for 280 bells.
The Mole Cricket featured in-game is similar in appearance to Gryllotalpa brachyptera, a mole cricket native to Australia, but it could belong to any of the Gryllotalpidae species.
Upon Donation to the Museum
In Animal Crossing
"My my my! Good Gracious! This is a frightfully large mole cricket, eh wot? Yes, rather large indeed. Interesting, isn't it? How its monotonous song drags on and on, while the insect is nowhere to be seen? That very secret is what sets the mole cricket apart from its relatives. You see, it lives underground, wot wot! It may be because it's become a bit scarce, but I believe most folk aren't aware the mole cricket exist. To be nitpicky, it's not actually a true cricket, but rather a distant relative of crickets and grasshoppers. It has ripplingly powerful forlegs for digging. In face, it's these mole-like limbs that give it its name. Its rear legs are shorter than a true cricket's as well. Bleeeeeech! All this talk of bug legs... Simply dreadful!"
In Wild World
When accepting the Mole Cricket, Blathers will impart his knowledge of the bug to the player:
"The mole cricket is a bug, of course, but it resembles a mole! Hence the name. It's clear once you inspect its lower half that it is most definitely a cricket. Blech! Add "imitating moles" to the list of unforgivable things bugs are capable of!
In City Folk
Blathers will say this when he reluctantly takes the bug:
"As both mole crickets and moles dig holes, it's said many parts of their bodies evolved on similar paths... But the mole is nowhere near as off-putting as this bug... How could they be? Moles don't have antennae."
In New Leaf
"Mole crickets live underground in tunnels they dig themselves, moving freely through soil. Their forelimbs resemble those of moles and are very suitable for digging. They have large wings that enable them to fly, but they can also swim if the need arises. In the past, people used to think the noise they made underground was actually coming from worms."
Mole crickets can be found throughout the world, specifically in more temperate climates. They are nocturnal and hibernate during the winter. They are rarely seen, as they live underground in tunnel networks, but are nonetheless regarded as pests. They have large forelimbs that can be used to dig and swim. They are roughly three to five centimeters long and are omnivores, meaning they eat both plants and animals. In some areas of Asia they are eaten, but only when fried. Mole Crickets also feature as a character in another of Nintendo's games - Mother 3.