My Space is Taken Up by Backups

A few months ago I had issues with my laptop and I sent it to the Apple store to be repaired. They recommended I begin to back up my laptop just in case, so at the store they helped me pick out a LaCie 500GB hard drive. One of the Apple guys set it up for me and backed up my laptop for me. He did something with Time Machine but I'm not sure what. Afterwards, when I checked the storage space on my laptop, a few GB were taken up, but not much. Shortly after this, I left camp for five weeks and didn't use my laptop.

I came back a few days ago and added some pictures and videos from camp to my laptop, so I decided to back it up again. I just plugged the hard drive in and hit the "back up" button, wherever that button was. Today I check my storage space. 61 GB were taken up by backups, and I had no idea how! I checked just a few hours later and:

How is that even possible?! Within only a few hours, I lost over 20 more GB to backups. I thought the backups were being stored on my hard drive, not on my laptop! I'm so confused right now. I went onto Time Machine and it says it's "On" but, honestly, I don't have any idea what Time Machine even is. I wanted to turn it off, because I felt that it has something to do with this, but if I turn it off will I lose all of my backups? And why are the backups on my laptop and not on my hard drive?

Thank you!