You cannot trim a clip in the media browser. To trim it, share it out and trim it in QuickTimePlayer, and then import it back in. That's the only way to get rid of the unwanted footage. Even trimming it in a project doesn't remove the unwanted footage, it only makes it invisible in that project. The trimmed portion still remains in the project although invisible. This is to allow you to restore the trim in case you made a mistake or want to make adjustments to the amount of footage you trimmed off. You couldn't do that if trimming permanently removed the footage.
So, after you have shared the clip to your desktop, open it in QuickTimePlayer. In the QuickTimePlayer menu click on View/Show Clips. A timeline will appear at the bottom of the screen. Click on the timeline to select it. Click on Edit/Trim. Move the right edge in the timeline to outline in yellow the portion of the video that you want to keep -- the rest will be removed. Click on the little Trim button in the lower right of the screen, to the right of the timeline. That will trim the clip to the portion that you outlined. Then Save and import back into iMovie. The trimmed portion is now permanently removed.
-- Rich