Adobe Premiere Pro - Select timeline region and remove it
This can be done very easily be setting In and Out points in the sequence timeline. 

In and Out points can be set using the I and O keys respectively.



Once you have made your select, press the Delete key to remove the selection. 

Make sure that the video and audio tracks you want deleted have 'track targeting' activated.



If you want to quickly remove the gap between all the clips, you can also right-click in the empty area and select 'Ripple Delete'.