If it's asking for FTP details, the problem is that Apache does not have write permissions to the files. The UX is pretty bad in that it doesn't let you know that it doesn't have write permissions, it just ignores that fact and presents the FTP details.
Here's where a lot of the security problems arise. As I mentioned, a lot of people are on a shared host and using mod_php. The only way they can make their files writable is to somehow give Apache write access to them. That opens up the possibility for other sites on the shared server to execute PHP that also writes to these files.
If you're stuck in this position, I'd recommend as a hack/workaround, to temporarily chown the files to www-data/apache. Perform the upgrade, then change ownership of the files back.
Ideally, if you're setting up your own server, I'd recommend FastCGI/SUExec:
http://www.howtoforge.com/how-to-set-up-apache2-with-mod_fcg...
You have much finer-grained control over how your PHP can execute. Alternatively, although I have no experience with it, you could try suphp to achieve a similar effect but by sticking with mod_php.