Personally, I was unconvinced I needed a DAC until I tried one. Being an Old Guy (tm), I come from a time when Good Audio did not require a DAC. (I did not own any good audio gear for a long time - having a family, then getting divorced, put my financial priorities elsewhere). When I became re-interested in Good Audio earlier this year, I immediately ran into the DAC issue. It was kind of funny how many things I had to read before I finally read something that explained what on earth a DAC was needed for.
Long story short: most music is stored in digital form these days. It needs to be converted to analog format prior to being amplified, and the digital to analog converter (DAC) does this. Many devices employ cheap DAC chips for this purpose. As you might imagine, cheap chips equal poor audio quality. Better chips (and better circuits) equal better audio quality. Devices like your iphone, a laptop, a tablet, etc employ really cheap DAC chips, that are very easy to improve upon.
The bottom of the scale for stand-alone DAC devices seems to be around $100 at present, and the Schiit Modi is absolutely the best DAC in that price range. You can get a better DAC, but you will spend a lot more money. There are a few cheaper DAC's, but honestly you probably don't want to go there.
There is also the legitimate question of how good your ears are. How much you need to spend on audio gear depends on how well your ears work. As an Old Guy (tm), I am a bit cautious about how much I spend on audio gear, simply because I know my old ears can't hear fine detail like they used to. So even if I was able/willing to drop Cubic Dollars on a DAC, if I could not hear the difference, then that's money wasted. However, if you're relatively young (e.g. < 30) and haven't otherwise done damage to your hearing, then you can probably go Hog Wild on a DAC and hear all sorts of Sonic Marvels. YMMV.
So all that being said, a Schiit Modi DAC is probably a very good first purchase. The first DAC I ever tried was a Modi. The difference between DAC and no DAC was dramatic, to me. Clarity. Less distortion. Most aspects of sound quality I could describe, improved. I was immediately sold on the necessity of a DAC (even with my Old Ears). Even an Old Guy like me can hear that level of improvement in sound quality.
So, yeah. I would recommend at least a Modi. Whether you need something better is a matter of opinion. Some folks preach that your audio budget should include the largest slice for a DAC. Other folks think you should get the best cans first, best DAC second, best amp third, and worry about stuff like cables, power conditioners, and left handed smoke shifters, if there's money left over.
