What does "doing something with a flourish" mean? Here is some context below.?

Doing it to attract the attention of others.

He's presenting the gift with a showy display, something that stands out and evokes emotion. Google the definition of flourish to understand it better. Here's the most relevant definition as per your example:

Flourish
noun
1. a bold or extravagant gesture or action, made especially to attract the attention of others.
"with a flourish, she ushered them inside"

showiness of some sort, like a bow and a wave of the one arm while offering the gift with the other hand like it was holding a platter of gold with the gift on it. Flourish pretty well refers to that arm wave, like signing your name with a flourish would mean you end it and kind of wave your arm into the air with the pen after you finish. If you ask someone to come in with a flourish, the idea would be that wave of your arm toward the inside of the room or house, "C'mon in".

The basic idea, though, is a use of the arms, head, body, to provide some floweriness, some show, a lot of emphasis, to whatever it is you are doing.

Making a big deal out of something simple or routine, meaning dramatizing or exaggerating whatever (words, reaction, expression, movement, etc).

An exaggerated movement.
Such as a very low bow, or a big wave of the arm when gesturing towards someone/something.