why the mariner killed the albatross?

Coleridge goes not give an exact psychological reason that the Mariner shoots the Albatross, and in truth, a simple explanation might work against the poem's metaphorical, poetic and elusive strengths. The bird is following the ship for a while, but the narrating Mariner does not express any emotional response to it. It seems clear that the motivation is meant to be unclear. If the albatross represents the "Christian soul," which the Mariner suggests a few stanzas above the shooting, perhaps this is Coleridge's observation of how we, as humans, sabotage our greatest assets for no discernible cause. The "soul" connection comes later when the souls of his dead passengers fly from the water and the Mariner compares them to the speed of his crossbow. Regardless, the Mariner's murder of the bird is meant to come off as impetuous and unjustified, an act he likely thought was of little consequence but which proved entirely otherwise.