Depends on the type of account, but here are some of the common methods of how this might happen:

• The attacker could be straight up guessing the password. (One possible way to mitigate this: the website can go "wow, 10 failed login attempts from that source. I'm going to ignore all attempts from there for 24 hours.")
• The attacker could be using previously exposed passwords. (One possible way to mitigate this: The websites should immediately require password reset for all users when that kind of data breach happens. For users: never use same password for multiple different services, certainly never reuse a compromised password even if it's for a different service. Also: haveibeenpwned.com)
• The attacker, currently using the same network, could hijack the session. (This was a really huge problem back in the day. In this day and age, websites should be using HTTPS, which limits this very much. Still possible if the site doesn't use HTTPS, and through some other vectors, e.g. malware or hijacked network hardware).

Also: Malware is a really scary big problem in that they're rarely targeting you specifically. Why do that, when they can million people at the same time and sift through that stolen data for most valuable stuff, right?