Once,there was a rabbit who was very proud of his swiftness. He thought that no one can could beat me him in the a race. One day he made fun of the turtle and said, "What a sluggish sluggard you are!" The turtle could not put up with this insult and invited the rabbit for to race him. The rabbit accepted the turtle’s invitation and the race soon started.soon. The turtle did not waste a minute and set off for his destination. The rabbit also run ran with his full speed and soon went far was far ahead from of the turtle. When he reached was near the winning point finish line, he thought, "The turtle is far behind; I should take a rest for a while. When he will reach near gets close, I shall reach the winning point finish line in a few leaps." To think After thinking this, he slept fell fast asleep. The turtle did not take a rest, but continued his struggle and reached the destination. When the rabbit got woke up, he ran as fast as he could, but when he reached the winning point. Seeing finish line and saw the turtle there, he was ashamed.

I don't need to correct your entry as there are excellent ones already.  However I do want you to change the title to "The Hare and the Tortoise".  This is the standard story from Aesop's Fables.  Also technically, a turtle is a sea going creature.  So a race between a land and a sea creature is impossible.  A tortoise is a land turtle.

Rabbit and Turtle

Once a rabbit was very proud of his swiftness. He thought that no one can beat me in the race. One day he made fun of turtle and said "What a sluggish you are!". The turtle could not put up with this insult and invited the rabbit for race. The rabbit accepted the turtle invitation and the race started soon. The turtle did not waste a minute and set off for destination. The rabbit also run with his full speed and soon went far ahead from the turtle.When he reached near the winning point, he thought " the turtle is far behind, I should take a rest for a while. When he will reach near, I shall reach the winning point in few leaps." To think this he slept fast asleep. The turtle did not take rest, continued his struggle and reached the destination. When the rabbit got up, he ran as fast as he could but when he reached the winning point. Seeing the turtle there, he was ashamed.

Rabbit and Turtle

Once there was a rabbit who was very proud of his swiftness. He thought that no one could beat him in a race. One day he made fun of a turtle and said "How sluggish you are!" The turtle could not put up with this insult and challenged the rabbit to race. The rabbit accepted the turtle's invitation and the race started soon. The turtle did not waste a minute and set off for the destination. The rabbit also ran at full speed and soon got far ahead of the turtle.When he arrived near the finish line, he thought "the turtle is far behind, I should take a rest for a while. When he nears, I shall reach the finish line in few leaps." Thinking this, he fell fast asleep. The turtle did not take rest, he continued his struggle and reached the destination. When the rabbit woke up, he ran as fast as he could but when he reached the finish line, he saw the turtle there, and he was ashamed.

(Great job! Keep writing! :)

Rabbit and Turtle

Once a rabbit was very proud of his swiftness (this is not necessarily wrong but very few people use this word. I think what you are trying to say is speed). He thought that no one can beat me him in the race. One day he made fun of the turtle and said "What a How sluggish you are!". The turtle could not put up with this insult and invited the rabbit for a race. The rabbit accepted the turtle's invitation and the race started soon. The turtle did not waste a minute and set off for the destination. The rabbit also run ran with his at ("with his full speed" is not necessarily wrong, but like above very few people use this phrase) full speed and soon went was far ahead from the turtle. When he reached near the winning point finish line, he thought, "The turtle is far behind, I should take a rest for a while. When he will reach near comes closer (will reach near is again not wrong, but is awkward phrasing), I shall reach the winning point finish line in a few leaps." To think this he slept fast asleep. After he thought this he fell fast asleep. The turtle did not take a rest, continued his struggle and reached the destination. When the rabbit got up, he ran as fast as he could but when he reached the winning point finish line (This is not a complete sentence because "but" is a conjugation. After a conjugation there needs to be an independent clause and "when he reached the finish line" is not an independent clause). Seeing the turtle there, he was ashamed.

Though there are a lot of markings, this is a very good paragraph. Most of the things I crossed out are not wrong but are uncommon phrases. Another common mistake you made was forgetting to add a pronoun or article like "a" or "the." This was very good and keep up the good work