1.1. ... at the beetle
=> in the general direction of the beetle
1.2. ... the dog snatched at the beetle
=> To make grasping or seizing motions
2.1. He snatched the paper.
=> To grasp or seize hastily, eagerly, or suddenly
snatch - Definitions from Dictionary.com
3.1. She caught my arm.
=> To catch (hastily, eagerly, or suddenly)
3.2. She caught at my arm.
=> To make grasping motions
Would you be kind enough to explain to me the mysterious “magic power” of the preposition “at” following transitive and intransitive verbs?