The software I have purchased is Dragon Speech Recognition Software. On another computer do I just copy the software to the memory stick and try to install it on the drive-less computer?
You should be able to simply copy and paste all the content from the DVD to the flash drive and then successfully install the software from the flash drive. The only time (generally speaking) that you run into a problem when doing this is if you're dealing with boot discs or copyrighted material, such as a movie or video game.
Let us know if a problem arises.