How does a share help in the discovery of a new block?
In order to find a Block, one needs to perform A LOT of tries - about 2^32 * the current difficulty. That number is so huge that solo miners rarely would be able to solve a block. A share on the other hand is a block of difficulty 1, which is at the moment a few million times easier to find.

Finding a share does not help in finding a block in any way - each hash is independent of all others. However, a share IS a proof of work done by the miner - they put in the effort to help the pool find the block. Once one miner finds a solution to the block problem, the pool earns money and can pay its miners - logically paying everyone for each share submitted. Those shares did not solve the block, but they were a proof that a miner was trying to solve a block. The more shares, the more work needed to be put.

Without shares, everyone would be essentially solo mining - there would be no way to prove that one put in the work, so there would be no reason to be paying the miners - they might be cheating after all.

A useful analogy would be lottery tickets - it is unlikely you will win, but if you live in a small town and decide to share the potential win with everyone that participates in lottery in proportion to how many tickets they bought, you would essentially have a mining pool. Every ticket, even if it does not win you anything, is worth a slice of the winnings - you had to pay for that ticket, so it's a proof that you put in the effort.