Why do I need to open an account with PayPal first?

This is mainly for your own security and is a common practice in just about all internet transactions done with reputable companies. By opening an account you create a unique identity complete with a password known only to you and PayPal. Opening the account is fast, free, once-only and doesn't commit you to anything, it merely ensures that, when you use their services, PayPal knows that it's you. Think of it as a key.
A great many people across the world already have a PayPal account and use it for Ebay auction payments or purchasing goods and services from individuals or companies who are otherwise too small to accept credit cards directly. Note that if you already have an account, you don't need to open another one to deal with us, just log on to the PayPal site with your usual access name and password.