Well, people could get a hold of your ip simply if you visit their webpage that contains an ip grabbing script. And there are also certain techniques which allow you to grab someones ip via signature links and images.
Many websites that register bots flag people based on patterns, for example if you make a lot of posts with advertising / external links, or if a lot of your posts are really similar. Also some website flag users as bots when they're using a proxy of some kind to make new posts.
A good way to get off the list would be to send the owners of the website an email explaining that you're not a bot and if they've got some sort of test in order for you to prove it. For the website that you've provided there is a contact page, I suggest you start there and write them a message: http://botscout.com/contact.htm
This is not the way to go around it. if you get denied of access.. then you have been denied for a reason. I have a script on all my sites where it will allow bots but if in anyway you try to inject it or flood it the ip automaticly gets banned. Only way around this would be to contact the webmaster. If that failes you could use a proxy, some sites do use a proxy blocker but most dont because they have to many flaws.
and just remember changing IP does not always work, its easy to detect if its the same person with a different IP then just ban them off again with the new one 😉