+1 Upgrades just do exactly what you would think they do. They increase the attack damage by a certain amount. For your marine it is one damage, so it then deals 7. The same goes for armor upgrades. A zergling with +1 armor will only take 6 damage when shot by a marine with +1 attack. Of course there might be other buff/debuff effects for example a Guardian Shield from a Sentry which is basically +2 armor.
While armor upgrades always decrease the damage taken by one, the exact effect of the attack upgrade depends on the unit. For marines each upgrade increases attack damage by one, but Ultralisks for example gain 3 attack damage per upgrade. (Thanks to @OrcJMR for pointing that our in a comment)
Those upgrades have a huge impact on the game. The player with the upgrade lead has a big advantage in fights, assuming that the army supply is around even.
The prime example it the matchup between Zealots and Zerglings. With no upgrades a Zealot needs 3 hits to kill a Zergling. With +1 attack the Zergling dies in 2 hits. If the Zerg player then get +1 armor a Zealot needs 3 hits again. The same logic applies for all other cases as well even though it might not be as impactful as in this example (but it often is).
Good players always try to attack the enemy when they have an upgrade lead. This is often refered to as hitting a timing. Hitting a timing when your +2 is finished but the opponent's is not can decide the fight in your favour.