Crossbows are ranged weapons. That means that on the attack roll you add your proficiency bonus and dexterity modifier, and on the damage roll you add just your dexterity modifier. Melee weapons are the same, except they use strength instead of dexterity. The two exceptions are finesse and thrown weapons. Finesse weapons use your choice of strength or dexterity (presumably whichever is higher, unless you’re not trying to kill your target), and thrown weapons use whatever modifier they’d get if not thrown (usually strength, unless they have finesse).
The base cases are page 196 in the PHB:
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 is Strength, and the ability modifier used for a ranged weapon attack is Dexterity. Weapons that have the finesse or thrown property break this rule.
Some spells also require an attack roll. The ability modifier used for a spell attack depends on the spellcasting ability of the spellcaster, as explained in chapter 10.
Proficiency Bonus. You add your proficiency bonus to your attack roll when you attack using a weapon with which you have proficiency, as well as when you attack with a spell.
The exceptions are back on page 146, regarding weapon properties:
Finesse. When making an attack with a finesse weapon, you use your choice of your Strength or Dexterity modifier for the attack and damage rolls. You must use the same modifier for both rolls.
Thrown. If a weapon has the thrown property, you can throw the weapon to make a ranged attack. If the weapon is a melee weapon, you use the same ability modifier for that attack roll and damage roll that you would use for a melee attack with the weapon. For example, if you throw a handaxe, you use your Strength, but if you throw a dagger, you can use either your Strength or your Dexterity, since the dagger has the finesse property.
For completion’s sake: Since he’s a Dragonborn, he has access to a third type of attack, a breath weapon, as described in the race stats (I assume you’re using the base PHB version, not the fancy EGW or UA versions, in which case it’s on page 32). This weapon works entirely differently; your sibling declares that he’s using the breath weapon, and then the creatures in range make a saving throw. As usual for saving-throw-based attacks, he rolls damage regardless, and the creatures take either full or half damage, depending on whether they passed or failed the DC (modified as appropriate to their vulnerabilities, immunities, resistances, or being a monk).