Though the Russian title literally translates to "The Dragonfly and the Ant", the insect depicted is a grasshopper; the word "стрекоза" was used for both species in the 19th century, when Krylov's adaptation was published.
Plot
The Ant is hard at work gathering food on a wheelbarrow, while the Grasshopper does nothing but play the fiddle, dance, and drink with its friend the Stag Beetle. At one point the Grasshopper even kicks the Ant away as it passes by the Grasshopper's table.
Winter comes and the Grasshopper and the Stag Beetle have nowhere to go nor anything to eat. The Stag Beetle dies. The Grasshopper begs the Ant to take it in and feed it, "just until spring". The Ant merely replies that when the Grasshopper is done crying, it can go dance some more. The Ant closes the door on the Grasshopper, who eventually dies from cold and hunger.