I'll try to break it down step-by-step as simple as I can, but no guarantees.
1) Download and install TI-Connect.
You've supposedly already downloaded it, now just install it. Do NOT plug in the USB cable before TI-Connect is fully installed.
2) Test to see if TI-Connect can detect your calculator.
You can do this by running TI-Connect and selecting an option in the menu (such as TI-DeviceExplorer or TI-ScreenCapture). Make sure you plug in the calculator and turn it on first. If everything installed successfully, TI-Connect should be able to detect your calculator. If not, uninstall it, delete Texas Instruments drivers, and repeat Step 1.
3) Download programs, applications, games, etc from http://www.ticalc.org/pub/83plus/ If the game you download comes in a .zip archive, extract it with the appropriate software. The actual game file should be one with a .8xp or .8xk extension (programs and applications, respectively) or something similar.
4) Now that it's extracted, just right-click on the .8xp/.8xk file and select "Send to TI Device" (which should have been added when TI-Connect installed). If that option isn't present, continue. If not, go to Step 6.
5) If there isn't a "Send to TI Device" option, then open up TI-Connect and select TI-DeviceExplorer. Go to Tools > Windows Explorer. That should open up the Windows Explorer file browser. Navigate to where the .8xp/.8xk file is located, and just drag to the TI-Device Explorer window.
6) Wait for the transfer to complete, and when it does, enjoy your game.
Additional Notes:
If the file was .8xp, it should appear under the [Prgm] menu.
If the file was .8xk, it should appear under the [Apps] menu.
If the file was an assembly program, then there are a couple ways it might be run:
a) Normal assembly program
Try this option first if you don't know what type of program it is. Hit [2nd][0] on your calculator to open the Catalog. Scroll down to Asm(. Select it. Now, hit [Prgm] and select the game you downloaded. It should look like Asm(prgmASMPRGM). If it's a normal assembly program, it should begin to run.
b) Assembly shell
If the above didn't work, then it probably needs to be run from an assembly shell. Most assembly shells will be apps (.8xk). Some popular shells include:
MirageOS: http://www.detachedsolutions.com/mirageos/
Ion: http://www.ticalc.org/archives/files/fileinfo/130/13058.html
Doors CS: http://dcs.cemetech.net/
CrunchyOS: http://www.ticalc.org/archives/files/fileinfo/325/32507.html
To send an assembly shell to your calculator, just follow the above steps. Also, if a game requires Ion, I recommend you choose a different shell (such as MirageOS or Doors CS), since those are capable of running Ion programs as well as their own.
*Phew*. Is that detailed enough? Good luck.