1. You need to re-size your TF2 layer in your scene to make it take up the full area. In OBS, click "Edit Scene", select your TF2 layer, and hit "Ctrl+F". "Ctrl+C" will also center a layer when in Edit mode.
   
2. 700 Max Bitrate is very low. It is not possible to get a crisp looking TF2 stream at that low of a bitrate.
   
3. Do not broadcast at 1080p with such a low bitrate. Scale your resolution way down, maybe 540p.
   
4. Do not broadcast at 60fps with such a low bitrate. Use 30fps.
   
5. Don't use High process priority unless you run into significant problem with Normal priority.
   
6. To compensate for your low max bitrate, you can try increasing your "x264 CPU Preset" to "faster" or "fast", but be warned that this may come with a significant performance hit and probably won't noticeably increase the quality. "Veryfast" is the best to use in 99% of situations.

This may also be useful in configuring your settings: http://obsproject.com/estimator

You can also use this analyzer to get more information on your stream settings: http://r-1.ch/analyzer/

A few things to sort out: First of all, I wouldn't recommend streaming at all with that set up.

I'd ask what your internet speed is but if you're lagging at anything above 800kb/s bitrate then it isn't high enough to stream. Also, streaming at 1080p 60fps requires somewhere around 3.5-4.5k bitrate depending on what kind of quality you're looking for and streaming 1080p 60fps with a 560ti is never going to happen, especially since if you only have a 560ti your CPU is probably fairly old and not that great anyway. To put it into context, I have a 3570k at 4.3ghz with a 7970, fairly top of the line and I wouldn't dream of trying to stream above 720p 60fps, and when I did stream (when it wasn't for VTV) I streamed at 720p 30fps to reduce lag further and that was with 3k bitrate on a 6mb/s upload connection. On a connection that lags above 700 bitrate I wouldn't try streaming at anything over 480p 30fps, if that. On that particular computer, I wouldn't stream at all because it will introduce very noticeable lag when playing. Also, streaming 1080p 60fps when you don't have quality options is very much not recommended as there are very few people, especially considering the size of the TF2 community who could actually watch it. Even 720p 60fps isn't recommended if your stream doesn't have quality options.
In short, your settings are completely out of balance and you're trying to stream with a computer and a connection that are entirely unsuitable for streaming.
If you do want to fix the letterboxing on your stream try this: right click the TF2 source in OBS, go to "position and size" and then "fit to screen".

Edit: after reading Lange's post, I'd agree that while it would be possible for you to stream, I'd recommend against it if you can't deal with stuttering and higher ping.

Thanks for the info! Will try all this and see how it works.

Thanks for the info. Also, my cpu is a i7-2600, and I've never noticed huge fps drops while streaming any game, it was always my ping that went up, no fps problems really.