This expression is quite common in spoken Italian. It is an idiom, which means that when it is translated literally it makes no damn sense. In fact, the literal translation would be "I recommend myself." Even in this age of rampant narcissism, this isn't going to fly.
Here is basically what it really means (although there is no equivalent phrase in English). It means to exhort, enjoin someone to do something. Often something that involves the persons it addresses taking care of themselves, or doing something that will be to their benefit or the benefit of both the speaker and the person addressed. Yes, I realize that's obscure. So here are some examples.
Mi raccomando, andate piano!
No direct translation, but expresses the exhortation that the person addressed take care in not speeding and that the person speaking has a stake in this, emotional or otherwise.
Studia, mi raccomando!
Teachers often say this to students to get them to study. It means that the outcome is important, not only to the student, but to the person speaking.
Siate puntuali, mi raccomando.
The speaker is telling the persons to be punctual, in a nice way.
It is not at all unusual for this phrase to be used all by its lonesome:
In this case, the speaker is expressing his or her concern that you do The Right Thing. What this might be can be understood from the context. But, as always, it binds the speaker and the speakee (just made that one up). I cannot hide my affection for this phrase.
For a further treatment which is even more obscure than mine, see what the Accademia della Crusca has to say about it. The Accademia is an ancient institution that studies the Italian language and attempts to monitor its purity (such as it is). In Italian.