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I was messing around with the settings hoping to remove the dots in the background to take a screenshot for a school project when I came across a setting called "Show grid" with a checkbox. I can't tell what changing this option does - nothing seems to change. Is it supposed to hide the background? Or does it do something else?

It does exactly that - removes the dots. You can also export image via file menu, which will export it without dots.

Thanks for the response! Something must be wrong with my program then because the dots don't go away when I uncheck it.
Also thank you for the info about the export option. Is there a way to increase the quality of the image exported? Some of the labels in my exported image are hard to read and I need them to be easy to read for my project submission. I think this is happening because of how "long" my circuit is (i.e. how far down in the editor it goes) but I don't think I can make it shorter because of what the project is (and because markers want it laid out a certain way).

Try to remove the check box and don't forget to click OK button to apply the changes. This feature was there from the very beginning and was always working.
No, you can't change the quality. You can try to export in .png format it may be a better quality.
Usually, big circuit can be broken to smaller pieces by creating sub circuits. Did you try menu Circuit/New Logical Circuit yet?