The modifier is directly linked to the stat, so a STR of 19 has a modifier of +4. That just happens automatically.
STR based attacks will therefore benefit from the higher modifier.
Skill checks will improve if they use the STR modifier as it has improved.
Basically, the modifier instantly increases to match the new total of the stat then that flows over into attack rolls, damage rolls and ability checks/saves that use that modifier.
From the SRD:
When a character makes an attack roll, the two most common modifiers to the roll are an ability modifier and the character's proficiency bonus. When a monster makes an attack roll, it uses whatever modifier is provided in its stat block.
Ability Modifier. The ability modifier used for a melee weapon attack