'Why do I log my workout on Daily Mile?' 'What is the purpose of having an account on DM and publishing how did you exercise today?' Most of the people who are not on DM have such questions, but I think most of the people on DM know the reason behind "logging your miles". Why? Keeps a track of your progress, when did you do what in the month probably? You see your realistic fitness goal for the month/year or something...it basically helps you...
Logging your exercise also motivates you. Not because others comment on your workout, but just seeing your past weeks miles makes you feel good about yourself. After becoming a part of DM, you can monitor yourself, what did you do which day, where did you go, which route you took to run last weekend etc. 
Logs are a part of an athlete's or a performer's life. If this athlete or performer is new to the field, logging helps a lot more. For instance, when I started running outdoor, logging my miles helped my gain some confidence in my outdoor jogs/runs because I myself could see some marked improvement. It is the best way to keep yourself motivated. Self-monitoring is much more rewarding than external motivation from peers or coaches because you can trust yourself! Yeah, but for this you have to be very true to yourself. Logging in correct miles at the correct time will help you become a better athlete. 
Similarly, preparing a PDP or a Personal Development Plan for your daily routine helps you as a person reflect on achieved goals for the day, or why some goals where not achieved. Even small things like diet can change if you prepare a small excel sheet planner for your daily diet. Keeping an account of food consumed and reflecting back in a week's time helps you monitor your sweet intake as well ;)
I can see changes in my own approach towards fitness and life because of logs. Many think it is a waste of time, but I feel it helps. But, one should understand only logging will not help, reflecting back on the logs and finding ways to improve the perceived flawed area or under achieved goals will certainly help you improve in you fitness and help you become a better athlete and a better person! 
Don't see DM as a way of showing off your daily workout, take the opportunity to get inspired by others, and also by yourself by seeing an improved you at the end of each week or month :)
Happy Exercising!
Cheers :)
