Cutting is probably the hardest part of editing. A first draft is quantity over quality, but all the preceding drafts are quality over quantity, which means that editing may involve cutting some of what you've written. If this task makes you think, "Hey, I worked really hard on this! I like it. No way am I cutting it!" here's how I've gotten through it:
The first story that I wrote for my college fiction writing class was much longer than what we were supposed to hand in. When I told my friend that I might have to cut a subplot, she advised me to keep a copy of the original story because I might want it for myself. Her advice taught me that this was only one story - if there wasn't room for all the subplots in this version, I could always use my original story another time.
I always polished the stories I brought to fiction writing class as if they were final drafts, which made it difficult to revise them. Sometimes revising meant barely touching the original documents. But whenever I was hesitant to slash what I had worked hard on, I remembered my friend's advice. My material was never gone forever. I could still keep my original drafts.
When I finished my first novel, I knew that I would need to edit if I wanted it to be a finished product. Using what I had learned from class, I slashed 50 pages from my first draft. It sounds hard, but here's what made it easier: First, I set my goal length higher than it needed to be so that I could cut pages without worrying about the book being too short (looking back, I should have set it even higher). Second, I saved a copy of the original that would stay as it was, while I edited a second copy. Second, I made a document entitled "cutting board," which was a temporary storage for pieces of the story that I cut but thought I might include somewhere else. Lots of sentences and paragraphs never left the cutting board, but knowing they would be saved somewhere made me more willing to cut them in the first place.
The point here is, it's okay to be attached to your writing, and it's okay to hold onto things just for yourself. When you have a hard time cutting, imagine that your story is a castle of Legos. When you build anything with Legos, you normally knock it down when you're done and then rebuild something new from the pieces. A story may sound better without all of the pieces you used, but keep in mind that you own those pieces. Whether you build the bricks yourself or steal them from a house you've always admired, they are yours. Whether or not you use every piece in one story, the pieces are yours to keep forever, until you build something new.