How a duke becomes a King.

 The Bishop of Bath and Wells

For Richard, Duke of Gloucester, to claim the throne for himself, he needed to prevent both Edward and his brother from accessing the throne. Fortunately for Richard, the Bishop of Bath and Wells  suddenly claimed that Edward V and his brother's were illegitimate offspring of Edward IV. Therefore neither of them had a right to the throne.

Dame Eleanor Butler

The Bishop claimed that, as a child, the future King, Edward IV had been pre-contracted to marry the daughter of the Earl of Shrewsbury, Dame Eleanor Butler.

Since this pre-contract of marriage was still in force when Edward suddenly and secretly married Elizabeth Woodville, it followed, according to canon (church) law, that the children of the marriage where in fact illegitimate!

How did the timing of this story help Richard claim the throne?

The timing of the story, as well as it's implausible nature, makes it clear that it was merely a device to bring Richard to the throne. Richard did not actually believe the illegitimacy claim but nevertheless he used it to usurp the throne. 

 A pre-contract marriage, after all, was not a marriage. Therefore, Edward IV was not commit bigamy when he married Elizabeth Woodville. 

Also, Dame Eleanor Butler had died before Edward's children were born, so she could not make either his marriage or children illegitimate.