I am running search campaigns for a while and i see in offer conditions for many offers in affiliate networks that "brand bidding" is prohibited in search ads.
The question is that do advertisers have some tools or software to understand which keywords i am bidding for their campaigns?
They do not need a specific software to monitor this, also some may have.
They just enter their brand name on a regular basis to make sure no one is bidding in front of them.
When a brand starts to get known, they get a certain amout of organic search traffic. They do not want an affiliate to put an ads on that traffic, which would easily convert, while they would have had that conversion for free on their main site.
Exactly as Jeremie said, those who really care about this rule are monitoring the key keywords to see whether someone is bidding on them. Once they see an AD, all they need is to click it to see whether it's an affiliate running these ads and by following the url, they can get the aff ID and get such affiliate banned.
Apart from wanting to keep the organic traffic to themselves, they also prohibit the activity in order to not let others to up the bid for related keywords, as many brand also run ads for their branded keywords. So basically, they want to sit on top of the organic SERPS and also make sure they are the only ones running paid brand-kw campaigns to get as cheap clicks as possible. On top of that, many have their brands trademarked, so google themselves might reject campaigns bidding on trademarked KWs.