Let's say you have a number like 35. This is the product of two prime numbers, and with a bit of calculation you can figure out that the primes are 5 and 7. Factoring is doing that calculation, and 5 and 7 are called the factors.

When the number that you start with is large (RSA-260 is 260 decimal digits) factoring is a difficult computational challenge. RSA-260 (and a bunch of other ones of various sizes) were published a few decades ago and someone just managed to factor RSA-260. One of the factors is 439732... (the number in the post body) and that lets you calculate the other one by long division (no big deal when you use a computer). Without reading the report I'm going to guess that factoring RSA-260 took some months on a cluster of fairly fast computers, i.e. a bit much for a random person with a laptop but within reach of government agencies or tech companies.

Factoring big numbers efficiently is a big important math problem that until recently was the basis of most web security. There is a scheme ("RSA") for using a composite number (like RSA-260) as an encryption key, where decryption requires knowing the factors, which are very hard to find. Typically the numbers encryption keys for web servers were 2048 bits which is around 600 decimal digits, so still out of reach for the methods and hardware that cracked RSA-260. But, for other theoretical reasons, the web has been moving away from the RSA scheme towards one based on elliptic curves (another math thing, don't worry about what it means exactly, it's a more sophisticated version of the same idea).

By which numbers it can be divided in to a whole integer.

For example, 10 has the factors 1, 2, 5 and 10.

Numbers that only have 1 and itself as factors, are "prime numbers"

It's explained at the top of the linked article.