First of all, try turning on your NDS with the DSTT inside but NO microSD card in it. You should get an icon that says "Menu?" in it. If you get that icon, that's a good sign. If you didn't get the "Menu?" icon, you'll need to return your defective DSTT to the store you got it from and exchange it for a new one.
OK, assuming you got that far...when you unzipped the firmware, did you get a TTMENU.DAT file and a folder named TTMENU with a bunch of other files and folders inside? Sometimes, they will be inside an "eng" folder, but you should NOT put the "eng" folder on your microSD card. Instead, you need to drop the TTMENU.DAT file and the TTMENU folder (with all its contents) onto your microSD card.
Then make a folder on your microSD card named "games" and drop 3 or 4 .nds files in it. When you view the microSD card, you should see the TTMENU.DAT file, the TTMENU folder, and the "games" folder.
Use the "safely remove hardware" utility before disconnecting the microSD card from your computer. With the DSTT in the NDS but with the NDS turned OFF, insert the microSD card into the DSTT. Turn on your NDS.
Then tell us what happens.
-Bri