I never understood the reason behind the practice of bowing before you step on the mat. I guess it's mainly because we don't do it at the gym I go to. Who or what are you actually bowing to? 
A simple courtesy and a small token of respect. Not a big deal either way. Some gyms recognize the Japanese roots and act on it. We bow on and off.
We don't require it, but a lot of people do it at our gym. I do it frequently but not always. I didn't come up in gyms that did it, but I like it. Basically I see it as saying "I take what we're doing here seriously and I am conscious of that every time I step on the mat." 
I don't bow to the mat when training. I do bow before entering the cage for an MMA fight.
Exactly what J Rockwell said... You leave the outside world outside when you step onto the mat. It is a tiny acknowledgement of the fact. You are telling your mind, "Now, I am training."
If you are wandering on & off the mat, taking cell phone calls or whatever, you're bringing the outside world onto the mats and that ruins the sanctity of the training space for everyone.
Also, you are not really bowing to the mat, per se, you are more bowing to the Dojo/training space which includes the instructors, other students and the art itself you are practicing. You are showing a token of humility for the space you are about to enter AND demonstrating a humble mindset, which is considered the only mindset appropriate for martial arts practice.
I come from a Traditional Martial Arts background myself so it just feels a little "wrong" NOT to bow when stepping onto a mat.
I bow out of respect for what the academy stands for & what training gives me back
I grew up doing judo, so bowing is par for the course.
I still do it, it's a sign of respect to the school and your training partners.
We don't bow normally but I certainly understand if thats tradition in some schools.
However I usually shake hands with instructor and after training with everyone out of respect to everyone. If I'm visiting other school I'll bow just in case usually :)
my back just hurts so it just looks like I'm bowing.
seriously though, it's a sign of respect for the space, dojo, instructors, and setting your mind right.
plus the dojo I train at has a strong kyokushin background.
I bow on and off every time, as well as my instructors and training partners. I have no tma background and don't consider it a tma practice, just respect.
We don't bow at my school but I always do when I travel and train at other schools just in case it is their custom to do so. When in Rome and all that
It's an acknowledgement that you're stepping into a situation where you or your training partners can be seriously injured. It's a sign of respect for what you're about to do and a reminder of the consequences if you're cavalier about it.
It's part of the art. Unfortunately Americans seem to hate art.
I came from a school in which we didn't bow. Eventually I started training at a school in which bowing was common practice. It bugged for the longest. However, as my "maturity" in the art has increased I've come to appreciate the gesture a lot.
bow...the mat brings out your character. respect the battlefield
Bowing is not about the mat. It's about you. Bowing before training always signified a willingness to be humble, patient, and open to correction in my years in the arts. In modern day terms, it's a physical representation of "leave your ego at the door."
I'm really not trying to disrespect anybody opinion here but I think that there are more tangible ways of showing respect. I.e. don't gloat when tapping someone, don't try and hurt your training partners, being quite and listening while the instructor is teaching, helping new students ect. Bowing I guess just seems a little karateish IMHO.
RocknRolla - bow...the mat brings out your character. respect the battlefield
well said fren
Showing respect to your training facility. Formerly, people viewed training as an honor. You were learning how to kill and to protect you relief. Personal ethics, behavior, and honor were huge whereas now everyone is too big to show respect and to strive to be a great human being. The goal now is just to look cool wearing your tapout gear and telling people you train.
It's a cultural difference between the Japanese and Americans. What we value as human beings and where we see ourselves in the grand scheme of universal culture.
Protect yourself *
It pays respect to anybody who has ever or will ever train on that mat.