For me, dramatic means getting into my NPCs and really thinking about how they sound and their mannerisms. I want the players to see at least a part of what’s in my head regarding this character. I’m currently running The Lost Mine of Phandelver for my sons and their friend and having a blast. I’ve run this adventure before and really enjoy how it holds the players’ hands for a bit and then lets them loose once they hit Phandalin, while at the same time giving the GM some tools to lead the players along.
As for the characters, you have a lot of options to introduce drama here and they’re great. The most obvious and inevitable one in Phandalin would be the Redbrands. They’re the local scum and villainy and they inevitably start a fight with the players, telling the player characters to leave their town and give up their stuff as well. I used this as a time to call the players names, poke fun at them, and introduce the idea that sometimes NPCs just aren’t nice. My sons’ friend pointed out that the ruffians were talking back and I had to explain that yes, they’re talking back and you can “talk back” to them as we’re just acting (drama if you wil) and then she got it—I just hope I don’t hear from her mother later.
In my mind, being dramatic doesn’t need to mean being out of control with the drama, but it should be plainly obvious what this person wants or doesn’t want. Players should quickly get the hint that they’re spoiling for a fight, fearful of something, itching to go somewhere, or desirous of something.
The flip side of all of this is to encourage your players to do the same thing! Let them be dramatic, let them tell you what they want and how they want to get it. Permit weird and odd approaches if you can, but try to be on the same page about what you’ll permit.