It looks like you have simple chat enabled. To change this go into you interface options and select the social tab. Once there it should be the option right at the bottom. Deselect Simple Chat and lock Chat. Once done, you can exit the settings and change things to your liking.
The only thing I can't understand is why would you want to remove the combat log?
Edit: I see you are playing a Nightelf Hunter. Here is a tip for you. On your pet abilities bar, there is a blue icon which when moused over says "Cower". Right now it's on autocast and this is exactly what you don't want. Right click the skill till the swirling outline stops. This indicates it is NOT on autocast. Your pet will not lose aggro (the enemies focus) and you can happily shoot away.
There is a bug (known by Blizzard) that all pet abilities that are NOT on the bar will automatically turn their autocast states to ON when you zone in and out. So everytime you see a loading screen, you will have to manually change it.
Good Hunting