Question: My instructions that came with my kit said that I should let the bottles sit for 3 days, then age in the fridge for 10 days. If I don't refrigerate, will I get bottle bombs?

Advice on forums and other places says to leave them in the bottle for two weeks or even upwards of a month, I doubt every brewer has enough fridge space to store 5+ gallons of beer for 1 or 2 months. So, I am inclined to think they won't blow, unless I bottle with too much unfermented sugar in them. 

Will leaving them out in room temp, cause them to pop?
Are there any other reasons for me to refrigerate my beer, aside from taste?

Answer: 
You'll only get bottle bombs if you over-prime or bottle too early (or have an infection). The difference between fridge, celler and even hot (80-100+°F) temps will not cause a normally-carbonated beer to explode, or it'd be happening commercially all the time.

That being said, heat and temp fluctuations aren't ideal for your beer.

Make sure fermentation is finished when you bottle. Don't over-prime. Be clean. Then just keep the bottles in the coldest darkest place you have access to.
As it is 2019, I would like to add a little nuance to this debate. 

I have left multiple beers to condition at room temp for 7 days after bottling, and then put them in the fridge. They were tasty, carbonated, and totally fine to drink. That said, letting them condition for an extra week or two has made some of the IPAs I have brewed improve. I usually bottle a few 355ml bottles so I can pop one in every few days past 7 days to see how things are going.

Lately I have been brewing some really hazy stuff. The IPAs I'm going for are heavily dry hopped, and this usually results in some of the hop particulate/plant matter being left over in the beer. I think these hazy IPAs are usually much better fresh, as this leftover hop matter can result in the beer not aging as well as some other ales. It may be the case that 7 days room temp conditioning + a few days/one week in the fridge will be better for this style. I have bottled a batch and it is conditioning right now. I will try and get back once I have tasted a few at various stages (1 week room temp, 2 weeks, etc.)