It verifies the amount of shares held by each shareholder on a schedule.
Not only this, but its in the blockchain FOREVER... not some spreadsheet that can just disappear or be changed. If someone wants to know how many shares I have, I can prove it by signing the div address... how would I prove ownership with a spreadsheet? Sure there are ways... but considering the block chain is available and can handle the function nicely, its a good use.
The percentage of the dividends received on your address is as good a proof as the sathoshis themselves, if not better, considering that they're not spam. And they're also forever on the blockchain. There's an easily done deterministic process to go from percentage of dividend received on a given time to amount of shares owned...