"A private key is nothing but a large pseudorandom number roughly between 1 and 2^256"
Not sure how the "pseudorandom" came in there - the private key might be random, or pseudo-random, or entirely deterministic - it's essential property is that it's just a 256 bit sequence that is unknown to others.
"It’s perfectly safe to give your Bitcoin addresses to other people, but extremely important to keep your private keys secret. "
The essential point trying to be made here is that the Bitcoin Address won't allow people to determine your private key. but there are a lot of reasons why you would not want to give your Bitcoin address to other people - the key reason being that it allows to track you. There is one philosophy that suggests you should change your Bitcoin address on every transaction.
Overall, though, a nice introductory article to Bitcoin - I'd recommend it to others.