Ship integrity is like a general health state of your ship. From the FAQ:
" Ship Integrity represents the overall wear and tear your ship has accrued whilst traversing the galaxy. It represents the over time stresses and strains put upon your ship, and requires a specific kind of engineering to diagnose and fix. If left unchecked, this will affect your ships overall hull strength.
The cost of repairing your ships integrity is based on the size of the hull and the amount of wear, so keeping tabs on it is advised to avoid a large bill. "
I aso read somewhere else that it affects the subsystems functionnalities as well, but found no confirmation of that.