Hi and welcome.
I'm a transmission user and it seems to work OK on my mac so I'll try to give you some help.
Here's what I would try, roughly in this order:
1. *restart the computer*. Often this will fix the problem.
2. *delete transmission's preferences file*. Sometimes the prefs files become corrupted (corrupted means that an error appears in the file, much like a mutation in DNA, and renders it inoperable). Prefs files are found in two places, either in
(your computer)/(your startup disc)/Users/(your account)/Library/Preferences
or
(your computer)/(your startup disc)/Library/Preferences.
For the ones in parentheses, substitute your own computer's particulars.
The first place stores preferences for your particular user account, and the latter for system-wide preferences.
There is a file called org.m0k.transmission.plist in my Library's preferences folder, and I suspect that this is the preference file for transmission. (sometimes they are named by the name of the developer, not the app). Delete the file by dragging it to the trash, and try restarting transmission. Transmission will regenerate a brand-new prefs file.
3. If that doesn't work, *delete transmission and reinstall* it. I like a little app called appzapper,
http://www.versiontracker.com/dyn/moreinfo/macosx/28764 which will find all the bits of the program that are installed and get rid of them. For reinstalling, make sure you have the latest version. It would also be good to have the latest version of os X, which is 10.5.7. The more up to date the software, the more bugs and incompatibilites will probably have been worked out by the developers.
4. After you reinstall it, but before you run it, do a *permissions repair* with Disk Utility, which is located in applications/utilities/ . Open DU, select your startup disk in the box on the left, and under "First Aid", select "repair disc permissions". The current mac OS is based on UNIX, which is designed to be a multiuser environment where not every user has access to everything. Hence the permissions files, which define which users have access to which files. If the permissions are screwed up, you might not be able to use a particular file or app correctly. Then after permissions are repaired, restart the computer, and see if transmission works.
5. Another thing to try at this point is to *repair the startup disc*. There are a couple of ways to do this. Probably the most straightforward way for the beginner is to reboot via your osX software install disc that come with your computer. You put the disc in the optical drive and restart; when you hear the "bong" you hold down the "C" key, and it should boot up from the disc. There is a copy of disc utility on the disk. Select your startup disk, and under First Aid again, click on "Repair Disk". This checks for errors that may have crept into the files and file structure. If it finds errors, repeat Repair Disk until no errors are noted. Let it do its thing and then restart from your usual startup disc and try transmission again.
6. At this point, if it still doesn't work, then there may be some other incompatibility between some other software you have installed and transmission. The way to check for this is to *create a "test" user account* that doesn't have any other added software in it. You do this in system preferences/accounts. Just create a new user, and then log out and log back in as the new "test" user, and see if transmission works. If it does, then some of that other software is the problem. You could try doing a search on the search engine of your choice to see if there are any known incompatibilites, or perhaps look on the developer's site. Another option is to add your other software, one app at a time, to the test account, and retry transmission each time until you find the culprit.
7. At this point if it still doesn't work, and you are still set on using transmission, you could try a couple of things: A *third party repair utility*, such as Disk Warrior or Tech Tool. Sometimes these will fix some problems that Disk Utility can't. If that doesn't work, as a last resort you could just *start over*, back up your files, and use Disk Utility to boot up, erase your hard drive, and reinstall OS X from scratch, and then transmission.
8. hope this helps-good luck.