It is not overly difficult. The biggest thing you need to be able to do is have confidence. Act as if you belong/ whatever the situation is and typically nobody will question you. Now, as far as the question "Who would be stupid enough to fall for that?" it's really quite easy. I will give a quick example: (please note, this is for educational purposes and shouldn't be attempted or copied if it would lead to any illegal activities)
There are applications that you can use to disguise your phone number, a notable one being Spoof App. So, lets just say you want to get a teacher's password at school (or you could just say an employee's password at a corporation... most of this process would probably work well in that context) (please remember, I am merely using this for educational purposes!). Well, you obviously already have the name of the teacher that you are trying to get a password for. Chances are it isn't that hard to obtain their username either, by some observation (or, for instance at my school usernames are determined by the first letter of your first name, followed by a period and then the last name). Then, you find the head of the IT department for the school, and hopefully a phone number to reach that department, or the head of it. You then fake the phone number, and call up the person of choice. Give them something like: "Yes Mr. John Doe it seems we are having a problem with your account down here at the IT department and Mr. (head of IT department) wanted me to take a look at it. Unfortunetly, he forgot to tell me your password. He's in a meeting right now and it would save me a lot of time and a headache if you could help me out."
It can be that easy. You just have to act like you belong, know the right names to mention and most of all, try to be charismatic. By mentioning something along the lines that they would be helping you out, you trigger an emotion within your target that makes them feel compelled to help you.
Please also note that this wasn't exactly an incredibly smart excuse for obtaining a password, but I was just trying to make a point.

