Hitman 2016 storyline

This is addressed in the intro cinematic (which plays after being cleared for field duty, when you pass the training tests).

During the training cinematics, it is mentioned that 47 has had prior training, and that his skills are off the charts. The man Diana talks to seems very apprehensive based on 47's past (or lack thereof)

The narrator (of the intro cinematic) speaks to 47, explaining that he was a force to be reckoned with as an assassin, and that through his actions, many men fell, but many also rose to power. There is an offhanded mention that 47 did not think about the effect of his actions.

When he tells Diana that "They called me 47", I was confused who called him 47 and why?

We see things from Diana's point of view: we don't know anything about 47. As the game says, it's like the Earth just spat him out.

"They" is whoever gave him the prior training that he clearly has.

As a personal interpretation: The barcode, having a number for a name, and his cold demeanor highly suggest that he's a lab-grown soldier (or at least a human who has been altered and had his memory erased). I've only cleared the training, not the actual missions yet, so I don't know if more is revealed about this.
Because the narrator (of the intro cinematic) mentions the men that rose to power due to 47's actions, it seems likely that he was initially playing for the other team, he worked for ICA's enemy.

Am I missing something here? Should I play other Hitman games like Absolution to understand its story?

No. It's a reimagining of the story. There is no game that explores what came before.