PayPal succeeded because they rode the eBay wave. Millions of people hit the site in its "online garage sale" phase, PayPal had three or four things that went for them:
* As a seller, you could provide an experience less terrible than mailing a postal order with a basically instant and free onboarding process (compared with a real gateway)
* As a buyer, it was typically easier and faster than postal orders, and they developed a reputation of being pushovers for fraud, which was important when dealing with "garage sale" tier merchants.
* They were very aggressive with signup bonuses to acquire market share. $5 for new account + $5 for referral.
* A lot of handwaving about the eventual revenue model. I recall wne it was "we'll survive on the float of customer balances, nobody will ever pay a credit card fee" and then a series of market shocks where they changed course-- all of which took place long after they had a critical market base.
This got them a major market presence-- to the point where I can recall Citibank being curbstomped with a similar product (c2it) despite even richer recruitment bonuses.
Eventually, they merged and sailed under eBay's flag and it was the default payment method there until quite recently. This created an :installed base" nobody else could compete with.
They then pushed their marketing towards "you don't have to give the merchant a card number" playing off security fears, but I suspect major secondary factors are "it's basically LastPass for payment methids" convenience (from before browsers offered to save card info) and that some people keep a PayPal balance as an "off the family books" way of financing their hobbies and such without the wife seeing your purchases on the statement.
Nobody really wants to give out details of their credit card (except maybe in the US) to random sites, and there is no alternative to PayPal.