Grit - Increases combat damage, both melee and long-range

Guts - Speeds up your individual crew's rate of healing, and increases their overall HP in combat

Insight - Increases frequency of updates to your Map

Spirit - Slows the rate of Sanity decrease

Graft - Increases kinetopede speed and fuel efficiancy

Guile - Increases your chance to dodge missiles in combat and successfully flee

Vigilance - Increases the chance your own missiles will hit

Esoterica - Increases the rate at which you gain Apprehensions, which can be spent to improve all other stats

If you're looking for the current level of your stats, they're the numbers under the upper crew member's portraits in the UI.

Obviously, beyond the stated descriptions above, stats are also used in stats checks throughout the game's narrative.