Am I correct in understanding that a Scrip Dividend involves the issue of new shares instead of the purchase of existing shares?
Yes. Instead of paying a cash dividend to shareholders, the company grants existing shareholders new shares at a previously determined price. This allows shareholders who join the program to obtain new shares without incurring transaction costs that would normally occur if they purchased these shares in the market.
Does this mean that if I don't join this program, my existing shares will be diluted every time a Scrip Dividend is paid?
Yes, because the number of shares has increased, so the relative percentage of shares in the company you hold will decrease if you opt-out of the program. The price of the existing shares will adjust so that the value of the company is essentially unchanged (similar to a stock split), but the number of outstanding shares has increased, so the relative weight of your shares declines if you opt out of the program.
What is the benefit to the company of issuing Scrip Dividends?
Companies may do this to conserve their cash reserves. Also, by issuing a scrip dividend, corporations could avoid the Advanced Corporation Tax (ACT) that they would normally pre-pay on their distributions. Since the abolition of the ACT in 1999, preserving cash reserves is the primary reason for a company to issue scrip dividends, as far as I know. Whether or not scrip dividends are actually a beneficial strategy for a company is debatable (this looks like a neat study, even though I've only skimmed it).
The issue may be beneficial to you, however, because you might receive a tax benefit. You can sell the scrip dividend in the market; the capital gain from this sale may fall below the annual tax-free allowance for capital gains, in which case you don't pay any capital gains tax on that amount. For a cash dividend, however, there isn't a minimum taxable amount, so you would owe dividend tax on the entire dividend (and may therefore pay more taxes on a cash dividend).