Yoroshiku o-negai shimasu よろしくお願いします. Japanese language is full of such “ready-to-use” expressions.
Each of these expressions should be used in certain situations and this one is a well-known example. Literally: “I beg your kindness”, or simply: “be nice with me”, “welcome me”, etc. Japanese people use it in various situations.
For example (and this is probably the meaning it had when I heard it for the first time), when introducing oneself. Japanese people are very much used to briefly introduce themselves, in what they call a jiko shoukai (自己紹介, “self-introduction”): telling your name, what you do and what you like (your hobbies, shumi 趣味), and inevitably ending by the consacrated locution: yoroshiku o-negai shimasu. “Please welcome me among you and be nice to me”.
Did you know it?
Some people will write “4649” for yoroshiku (yo[n]-ro[ku]-shi-k[y]u). And deliquents (bosozoku 暴走族) like to write it phonetically in kanji. For example: 四路死苦 …
A completely different situation where the same expression is used is when formulating a request or asking a favor. For example: “Please don’t use you cellular phone on the train … yoroshiku o-negai shimasu … please!”
This expression is thus very useful and I really recommend everyone to remember it, because it is widely used. Finally, let me also mention the “familiar” version: yoroshiku よろしく, which is used among friends or family. For example: “Honey, could you help me carry my bags? Yoroshiku!” (=”Too kind, honey, I knew I could count on you!”)