Before you get into sound quality the more important thing to know is that, unless you're Tank or Neo on the Nebuchadnezzar's multi-monitor rig and can read 1's and 0's and hear "oh there's a drum hit, there's another one, and now here comes the guitar solo...sweeeeeeeet" like when Tank went "I see blonde, red head...", the DAC is an absolutely necessary chip to have because you can't hear 1's and 0's at all. The DAC interprets that into an electrical signal, which is eventually amplified into a stronger electrical signal that can move transducers like speakers or headphones, which in turn move air to make the sound you can hear.
As for variances in quality, that has a lot more to with how the signal is handled after the DAC chip. Lower channel separation can mean narrower soundstage (not that one DAC rated for 110dB would be audibly worse than one rated at 125dB), in some cases you can perceive slightly louder as significantly better so some DACs and CDPs have line out signals stronger than 2v. While a 2v DAC complying with Sony and Philips standard might lose out to a sneakily designed DAC with 2.2v on a subjective test that doesn't measure for this at the start, I've owned a DAC that outputs 6.5v when USB is used and it sucked.
It improves the sound in that it is a necessary component, you can't make audio from a digital source without a DAC.
As for how they work, a DAC is almost always created as an integrated circuit. It contains circuit blocks which decode a digital signal and converts it into analog. There are various architectures which are used for the conversion such as delta-sigma or R2R, you can research these or ask a follow up question if you want to know more specifically how they work.
In reality, no DAC we make can perfectly create the analog signal that is intended, they will all have some limitations and tradeoffs with noise, distortion, frequency response, linearity, resolution, crosstalk, power, size, cost, manufacturing process, etc. However, technology today has advanced far enough that even a cheap DAC can perform very well. You tend to get less for your money upgrading the DAC than you do speakers or headphones.
Very open ended question. We have a hard time answering it without knowing your background knowledge. Do you know nothing of electronics. Do you understand a little bit or where are you starting from here. And are you wishing to learn how conversion from digital to analog occurs in detail? Or do you wish to merely get an idea why it is done this way?
Look at the following links. If it doesn't click after puzzling over it, then skip it and try another. I think if you learn how an R2R DAC works that is the best approach for getting the concept. Even though few DACs are R2R ladder DACs the ideas help you then understand other ways to make a DAC. That and try to formulate a more detailed question regarding what it is you are hoping to learn.
You can also read these two Benchmark blog posts by John Siau on the ESS 9018: