What does this mean: "I'll be with you in a minute"? Does it mean "I'll call you after one minute" or "I'll contact you after one minute" or something else?
Well usually this is used when speaking face-to-face, and is taken to mean
I'm occupied right now, and I'll speak to you in some (short period of) time.
Not exactly one minute, but a short enough time. Similar usage to "Just a second" or "Gimme a sec"
If on the phone, it should probably mean he'll call back after a minute or so.
As @Martha says - On the phone, this would likely mean you should hold. The correct statement if he intends to ring you back would be
"I'll get back to you in a minute".
As JoseK said, this is most often used in person, and it means something like "hold on, let me finish this, and then I'll be able to help you/talk to you."
If someone said this to me on the phone, I would definitely interpret it as "please hold", not "I'll call you back", and most certainly not "Please call back in a minute".