Interesting

No legitimate websites will store passwords as plaintext, instead their salted (mathematically modified) hashes. They do not have your plaintext password but there is a mechanism of checking your password attempt. If the hacker interrupts this mechanism, they can steal plaintext passwords every time they are used. However, most of the time they "just" gain access to the database and get the hashed & salted versions, and can often find out what the hash & salt algorithm is by reverse-engineering their own entries. Then, they obtain a list of top 1B+ common passwords (includes all 4-digit numbers, "password" and curse words in all languages, pretty much every English word imaginable, most given names in the world, every date from the last 100 years in various formats, correct horse battery staple and entries from previous breaches) and for every hash & salt they compute, they check if it matches any known hash & salt of any user they're interested in (could be just a few or all) matches. This dictionary attack will match the weakest passwords, and any matches give the hacker confidence that their method is correct. After that, they either start extending the dictionary (combining words with other words, letters and numbers), or brute forcing every combination of numbers, letters and symbols (with some heuristics: most passwords use way fewer symbols than letters).

So "password breaches" are concerning because the hacker gets any number of retries for anyone's password if they have the time and computing power. Therefore, it is very advisable to change your password after such a breach is discovered. This takes anywhere between 1 day to a month so if your password cannot be cracked in that time, you should be reasonably safe.