Start with the usual equations for dissolution of carbon dioxide in water:
CO<sub>2 (g)</sub> <---> CO<sub>2 (aq)</sub>
CO<sub>2 (aq)</sub> + H<sub>2</sub>O<sub> (l)</sub> <---> H<sub>2</sub>CO<sub>3 (aq)</sub>
H<sub>2</sub>CO<sub>3 (aq)</sub> + H<sub>2</sub>O<sub> (l)</sub> <---> H<sub>3</sub>O<sup>+</sup><sub>(aq)</sub> + HCO<sub>3</sub><sup>-</sup><sub>(aq)</sub>
In addition, the carbonic acid can react directly with the ammonia present
NH<sub>3 (aq)</sub> + H<sub>2</sub>CO<sub>3 (aq)</sub> <---> NH<sub>4</sub><sup>+</sup><sub>(aq)</sub> + HCO<sub>3</sub><sup>-</sup><sub>(aq)</sub>
And also, the ammonia will react with the hydronium produced by the carbonic acid:
NH<sub>3 (aq)</sub> + H<sub>3</sub>O<sup>+</sup><sub>(aq)</sub> <---> NH<sub>4</sub><sup>+</sup><sub>(aq)</sub> + H<sub>2</sub>O<sub>(l)</sub>
More CO<sub>2</sub> will dissolve in ammonia because these last two reactions cause the second and third reactions to move right (by applying Le Chatelier's Principle), and thus the first reaction also moves right (Le Chatelier's Principle), thereby increasing the solubility of carbon dioxide.