How am I supposed to update my software when the download file is so large?

You need to improve your space management on your startup drive because you need more than 15GBs for an OS upgrade. Less space is needed for an update which is a smaller file. In general you should maintain a minimum of 20GBs of free space, and at least 30GBs when doing an OS update/upgrade. The extra space needed for an update/upgrade is temporary because it is only needed during the update/upgrade.

How to Free Up Space on The Hard Drive

1. You can remove data from your Home folder except for the /Home/Library/ folder.
2. Visit The XLab FAQs and read the FAQ on freeing up space on your hard drive.
3. Also, see Freeing space on your Mac OS X startup disk.
4. What is 'Other' and 'Purgeable' in About This Mac?
5. Files That Make Up the 'Other' Storage Category, and How to Remove Them
6. Free up storage space on your Mac.
7. See Where did my Disk Space go?.
8. Be sure to Empty the Trash to recover the space.
9. Replace the drive with a larger one. Check out OWC for drives, tutorials, and toolkits.
10. Use OmniDiskSweeper or GrandPerspective to search your drive for large files and where they are located.

I do wonder why is it always Apple's fault? May I suggest, in addition to the fine suggestions already offered here, that you might simply purchase an external drive and transfer most of your files to it? These days such drives are fairly inexpensive, easy to install and can do the job. Best of luck.