I have always used the sana/sene prefix as a sort of exclamation.

Like eg. "buraya gelsene" = come here! and "buraya gel" come here (softer)

So I would say that bana bak is look at me! It's more informal than bana bakınız.

-sene/-sana is actually
-sen + a (vocative particle)
-sen alone can be used to make a suggestion. It's actually wish-condition suffix.

gelsen = I wish you come or what if you come etc.

gelsen ya = why don't you come or I think you better come or something like that.

"-sen a" or "-sen ya" usually becomes -sene/-sana.

It's the 2nd person singular form. Other forms would be:

gitsem ya = I think I better go
gitsene (< gitsen ya) = what don't you go, I think you better go, I suggest you go, etc.
gitse ya = I think s/he better go, I suggest s/he go, etc.
gitsek ya = I think we better go, I suggest we go, etc
gitsenize (< gitseniz ya) = what don't you go, I think you better go
gitseler ya = I think they better go, I suggest they go, etc.

Aslan2's explanation is exactly the one I had at school when we were taught about the Şart kipi. I'd just like to add that İ see it this way:

Bana bak! - EMİR
Baksana bana! - TATLI EMİR.

Usually you give an emir more angry, because if you werent angry or dissatisfied you would use rica or geniş zaman (Ayran verir misin? Ayran verebilir misin? Ayran ver! Ayran versene!). I think you could use 'tatlı emir' indeed just to express you want someone to do something for you, without being rude or angry.
Also, it may express something like keşke.. like the example Erdinç gave about the traffic.. you wish the person did as you 'emired', but he does not.

Bana çikolata getirsene

İ wish you share it with him then..lol

Btw,who is singing this song ?

Will search it then.
Çok Tşk