I assume that you're talking about the +1 that you're seeing on certain weapons in the starter set.
These are magic weapons. This +1 is indeed applied to the attack roll and the damage roll. A magic weapon may also have other properties (like the axe in Thundertree that has advantage against plants).
Unless, you're referring to the +4/5/7 on the character sheets. This only applies to the attack roll. A weapon attack is represented with two different expressions. Here's the Greatsword from the Ranged Fighter:
Greatsword +4 2d6+2 slashing
I'll break that apart now. The first entry is the weapon type. You can by different weapons, this has two effects, if you're proficient, your attack modifier only changes if it's a finesse weapon and you decide to use Dex for the attack instead of Str. But your attack roll is your proficiency modifier (+2 at L1) plus your ability score (the ranged fighter's starting Strenght is 14 so the modifier is +2).
The second entry is the attack bonus, we just calculated that above. It's determined by whether or not you are proficient (the fighter is proficient with all weapons, other classes less so, see the box with "proficiencies" listed for the weapons you can add this bonus to). Then you add the appropriate ability modifier. Melee attacks use Strength by default, but finesse weapons can use Dex (not required). Ranged weapons use Dexterity by defualt, but thrown weapons can use Strength (not required).
Finally you have the damage. This is the damage dice indicated by the weapon with the ability modifier you're using for the attack added to it (so the Greatsword has 2d6 from the sword and +2 from your Strength modifier).