Why does a mix need headroom and dynamic range? In one sentence, it gives the mastering engineer more room to work with.
If I compared a hair stylist to a mastering engineer, hair length would be mix headroom. If someone came in with 18" of hair, the skies the limit as to how she could style it. But, if they came in with only 1" of hair, her style choices (options) are very limit. In audio mastering, no headroom limits your options.
In the audio mastering process, a series of EQ boosts and cuts are performed. Most of the time you're going to need to boost something, even if it's only a little +2db boost at 100hz. Well, if the song is already at 0 volume level or higher, you might not be able to make a necessary boost without distorting.
And if a song has very low dynamic range (the meter barely moves) it's probably over-compressed. Which means it could lack punch, power, clarity, or could even limit EQ options.
I want a song mix with some headroom and decent dynamic range. "I" want to EQ it as necessary, "I" want to compress it as necessary, "I" want to be able to set the overall volume as necessary, and I don't want to work with a distorted mix.
This section actually ties in with a previous one where I talk about not giving the mastering engineer a song that's already 75% mastered by you. Don't do it! And if you do, just master the other 25% yourself and save your money.