The number under the "To hit" column on the character sheet, (and followed by the phrase "to hit" in monster blocks). Represents what we call the "To hit modifier".
To use this, when you make an attack (melee or spell), you roll 1d20 and then add the modifier to the result. You compare this total to the target's AC and if it's greater than equal, you have a hit.
Then you roll the damage dice (1d8+1 or whatever). For monsters in the starter set, they also give you the average of this result in case you'd rather not roll the dice and just use the average.
Certain spells are different and require a saving throw for the monster. For this you need two different things. You need the caster's spell save DC (this is listed in the far right column of the starter character sheets or in the monster's state block). The target(s) must then roll a saving throw and add the appropriate modifier (usually dex). If they have proficiency they can also add their proficiency bonus (character save bonuses are precalculated in the "saving throws" box on the character sheet, and when a monster gets proficiency it's listed in a "saves" line).
Hopefully this information will help you resolve attacks in 5e.