How to select the correct Resolution/Aspect Ratio
Black Bars are always caused by an aspect ratio mismatch. Facebook uses 16:9 while ResQ's display for example uses 16:10 if I am not mistaken.
The easiest but definitely NEVER recommended solution would be to stretch your input to meet the desired output aspect ratio. This would negatively affect your image quality and the shown proportions on your stream, so please do not do this.
The better solution is to set OBS to a 16:9 resolution to meet Facebook's requirements. Now in OBS you can fill the black area with a graphic, follower text's, useful information or whatever you want. As a result, your viewers will be able to watch your stream on their 16:9 display without black borders and you do not have to stretch your input and screw its proportions.
So awbix is correct, set your base and output res in OBS to 1280x720 if you want to stream at this resolution. Add your sources to OBS and if you capture your Monitor for example you should see the black borders in OBS. Move your source to the bottom or right for example and fill the black area at the top or left with the mentioned graphics and text's. The Facebook webplayer should then show no black borders for most of your viewers. If you view the stream in fullscreen on a 16:10 or 4:3 display you will of course see black borders though. But generally most of your viewers will view it on a 16:9 display and have no black borders.
