The symbol of the bear and the staff, goes back to the early middle ages and probably has its origin in entertainments, including bear baiting, put on by the nobility for important occasions. It is the symbol for the Warwick family and the famous king maker himself, Richard Neville and was on the livery of all his retainers which numbered many thousands. They fought with him in the Wars of the Roses. Its been the symbol of Warwick and Warwickshire at least 700 years.
We know that there is a link between the symbol of the bear and the staff with Warwickshire's most famous son, Shakespeare, because if figures in one of his plays, The Winter's Tale, where the stage direction is 'exit pursued by a bear'. An interesting point is that I'm not sure whether the bears were actually indigenous to England in the middle ages. probably unlikely.

Here's one possible answer: http://www.rugbyadvertiser.co.uk/news/local-news/so-why-is-warwickshire-associated-with-bears-the-most-googled-questions-revealed-1-6742021