So, how exactly do I "observe" a leader?

[e11]

All I know is that after cutting one of these goobers down I get told how many more killings or observations I have left before leveling up.

What does that mean?

[MASTAN]

Approach him in stealth and when close enough(several meters) there'll be a button prompt to hold.

[Alexoferith]

The distance is actually quite generous. However, in certain case, the Dev. put a big dude right where you need to go (when you are still early stage, i.e. no grappling hook).

[e11]

And why not just kill him when I'm that close anyway?

[MASTAN]

You get additional point for stances upgrade per observation and can get all 4 basic stances faster. Nothing more.