Theory:

At last by starvation and famine made bold,
All dripping with wet, and all trembling with cold,
Away he set off to a miserly ant,
To see if, to keep him alive, he would grant
     Him shelter from rain,
     And a mouthful of grain.
     He wished only to borrow;
     He’d repay it tomorrow;
If not, he must die of starvation and sorrow.
Explanation:
  
The cricket had not saved any food for the future, and he wondered what will happen to him if he did not have food. He would have to starve without food if he did not do anything now. He was soaked wet and shaking because of the extreme cold weather. He gathered courage from the fear of being without food; he started his journey to visit the ant and ask his help. The cricket wanted to borrow from the ant and repay it in the future (tomorrow refers to the future, not the next day)
  • a staying place for the winter
  • a mouthful of grain (the quantity of grain that is sufficient for insects like cricket)
If the ant does not give him food and shelter, the cricket will die of lack of food and disappointment. The poet refers to the ant as "miserly" here. Miser refers to people to save a lot and spend very little. It can be given a positive meaning here, to praise the ant's quality of thinking about the future and preserving food for the cold days.
 
Meanings of difficult words:
  
Words
Meanings
starvationsuffering or death due to lack of food.
faminescarcity of food.
drippingso wet as to shed drops of liquid.
miserlya person who hoards wealth and spends as little money as possible.
 
Rhyming words:
 
  1. bold - cold
  2. ant - grant
  3. rain - grain
  4. borrow - tomorrow - sorrow
Reference:
National Council of Educational Research and Training (2008). Honeydew. The Ant and the Cricket - adapted from Aesop's Fables (pp. 21-23). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.