i don't understand why apps need to control my computer

It's Apple's fixation on security. They don't want to be responsible for anyone getting malware, viruses, or hacked, so they're lockiing down the OS. See http://support.apple.com/kb/HT6026 and http://support.apple.com/kb/HT5914 for starters.

I categorically deny access for anything and haven't had a problem because of it.

Other than Skype (which definitely does not need to use the Accessibility API), I don't know what any of those apps is. If you don't know why the access was granted, you should remove it. It's very security-sensitive, because any process that has access could potentially function as a keylogger, or could do pretty much anything else.

syberknight wrote:

the statement "allow the apps below to control your computer" kinda freaks me out ;-)

Then it is doing its job as designed. 🙂 Always think twice before handing over your admin password. You are giving that developer full control over your machine and all of your data.

The assistive devices option is in the same category. Even if they don't ask for your password directly, they are taking such low-level control of your machine that they effectively don't need to ask for your password anymore. They can just take it.

To be fair, these apps are almost certainly not absuing this privilege they are asking for. They are just trying to do some low-level system magic that is normally not possible. Usually such abuse is discovered eventually.

This has just happened to me with Dragon software. That's why I am searching here for someone to tell me How I run the software without letting it "take over" my computer.

You might want to consider starting a new discussion. Since this one is a couple of years old, less people are likely to look at it. A new post would be much more visible. You can link to this one.