Check this out it's a chart
http://www.apassion4jazz.net/jazz-chords-scales.html
You can see that different scales will fit different chords, because they share some of the same notes. The mixolydian fits a V7 chord and a few others (see under dominant chords). When you start to encounter these chords the Mixolydian will be useful because it will quickly allow you to improvise whilst staying in harmony (rather than creating dissonance).
V7 chords occur quite a lot so the Mixolydian is quite useful for this. If you play a major scale over a V7 chord, then the major seven will clash with the minor seven of the chord, so it would probably be better to play mixolydian instead. It's that simple.
The second use of the Mixolydian is just for it's sound. Different modes have different sounds or moods, that's the point of using them. To me, the mixolydian sounds quite bouncy, 'cheeky' and very good for punchy rhythm. That's just my take on it but essentially different modes have different moods so the easiest way to find out what it means to you is to improvise with it and see how that flat 7 works for you.
To improve your ear, I think the best way is just to play things back. Start simple and go from there. Don't be afraid to use Tab sites and song sheets and things like that to get a bit of assistance at first because this will help you to get a feel for common patterns and progressions quicker. Start by finding the root note, but if you have trouble identifying that, then seriously just cheat at first and look the song up online to get the chords. This will help you to hear the root note and you'll be more able to look for it next time round. There's always opportunities to do it without assistance in the future but get your bearings at first.
Also you can download programs that train your ear to hear intervals, or 'interval training' which is very very useful for quickly identifying jumps in pitch. Have a look at
www.download.com and type in ear trainer or similar. Those things really help.