You have no people. You are no longer my son.
This is one of the first lines of dialogue in the comic; it sets the tone for the rest of it. T'Challa/The Black Panther has been ostracized and cast out by his own family -- his father, specifically, whom he had a special relationship with. This makes T'Challa very upset and a bit angry; it will also foreshadow his actions later in the story in terms of his aggression/violence and T'Challa's wanting to bring Wakanda together rather than to tear it apart.
And so there is war.
This is vital to the development of the comic. At this point, the main conflict of the comic book is illuminated and Black Panther's aggressive attitude that was foreshadowed earlier by the Panther's conversation with his father is reinforced.
My name is my nature. I can track a body through wind and rain, for I track not the body, but the soul within.
On the surface, this is a very cool and engrossing quote. It tells readers the various and immense powers of the Black Panther. It also highlights how evil the soul of the woman who encouraged/instigated the hateful uprising is and how the Panther will try his hardest to stop her.
Call the soldiers back, my king! We must not massacre our own people!
This was said by the soldier to try to curb his King's (T'Challa) aggression and prevent his approval rating (or its equivalent in Wakanda) from plummeting down even further. T'Challa ends up taking his soldier's advice, ensuring no further carnage and allowing the citizens to get out of the woman's hateful spell sooner.