In the 5e Player’s Handbook, the Wizard Class on page 114 defines a spell attack modifier, but the spells I have looked at all specify attack damage without using this modifier or mentioning it. I can’t determine what it has to do with anything. Where does it come into the game?
Spellcasting Ability
Intelligence is your spellcasting ability for your wizard spells, since you learn your spells through dedicated study and memorization. You use your Intelligence whenever a spell refers to your spellcasting ability. In addition, you use your Intelligence modifier when setting the saving throw DC for a wizard spell you cast and when making an attack roll with one.
Spell save DC = 8 + your proficiency bonus + your Intelligence modifier
Spell attack modifier = your proficiency bonus + your Intelligence modifier
The attack modifier is used when making your attack roll. This is the roll to determine if your (spell) attack hits.
The process is the same as for other forms of attack:
Roll d20, add your attack modifier
If the result is equal or higher than the target’s AC (Armour Class), then you have hit.
If you hit, you roll for damage. If you miss… you don’t.
(Note, some spells do not require an attack roll. Refer to the individual spell descriptions.)