You're going to need a video editing program to do this. Pinnacle Studio is a reasonably-priced editing suite that can do this (pretty much any video editor can, so you may try some free ones first.)

the trick is that the video has to be in a format that the editing suite can handle. If your editor can't handle the DVD's format (VOB files,) you can use something like handbrake to convert the DVD stuff into and AVI or MPEG file that the editor does know how to handle.

Beware: Ebvery time you change the video's format, the quality degrades a little bit as it gets uncompressed and re-compressed.

Your work flow is going to look something like this:

1) Rip the DVD using AnyDVD.
2) Convert the DVD to a video file using Handbrake (optional, but probably necessary.
3) Use a video editor to cut out the scene you need and save it to disk.