You are proficient with your unarmed strikes, which deal 1d4 slashing damage on a hit.
Do talons just deal 1d4, or is the relevant ability modifier added too?
You add your Strength bonus to any melee damage you do unless a rule specifically states otherwise.
As per the description of the Strength ability score on page 176 of the PHB:
You add your Strength modifier to your attack roll and your damage roll when attacking with a melee weapon...
This general rule is always in effect unless it is explicitly overruled by something else. For instance, a weapon with the Finesse weapon special property states that you may instead choose whether to use your Strength bonus or your Dexterity bonus. Generally weapons only describe their base damage and the normal melee/ranged strength/dexterity bonus is assumed.
The rule for weapon damage states:
When attacking with a weapon, you add your ability modifier--the same modifier used for the attack roll--to the damage.
It is important to note that unarmed strikes are not weapons though they are used to make melee weapon attacks. If they're not weapons, one might think they don't obey the rule stating you add your ability modifier to weapon attacks.
However, in the 2015 errata that removed unarmed strikes from the weapon table, the writers also baked the general rule for adding the ability modifier into the rule for unarmed strike damage directly:
On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier.
Page 10 of the official Sage Advice Compendium has since been updated to confirm that all melee weapon attacks (ie, regardless of whether they are made with a weapon or an unarmed strike) have their attack and damage rolls modified by an ability stat modifier (strength by default).
An aarakocra's talons are used to make unarmed strikes but they have the unique quality of using 1d4 instead of just 1 for the base damage. The strength modifier is then added to the base damage of the attack as normal for 1d4 + strength mod net damage.
Optionally, the Aarakocra, like many small but quick monsters, is listed as a monster with its Dexterity modifying its claw attacks, i.e. its natural attack is effectively a finesse weapon. Either way, the relevant ability modifier is added to the attack. Check with your DM as to whether that is Strength, or Dexterity, or your choice.
The rule that normally adds ability modifiers to damage rolls, as stated on page 196 of the PHB, reads (emphasis as in original):
When attacking with a weapon, you add your ability modifier — the same modifier used for the attack roll — to the damage.
As stated numerous times in the Sage Advice Compendium, unarmed strikes are not weapons. (They are melee weapon attacks, but that's not relevant here.) Thus, no ability modifier is added to an Aarakocra's unarmed strikes.
Note: I'm not 100% clear on the timeline, but it appears that, when the Aarakocra race was first published in the EEPC, the PHB still classified unarmed strikes as weapons that dealt 1 damage. If this is true, then, at the time, an Aarakocra's talons would have dealt 1d4 + Str mod damage.
However, when errata later removed the unarmed strike from the weapons table and defined its damage to be 1 + Str mod, the Aarakocra was apparently overlooked, and so its Talons ability changed from replacing 1 with 1d4 to replacing 1 + Str mod with 1d4.
In the absense of any errata updating the race, there is no reason to copy the Strength modifier from the unarmed strike's default damage and apply it to the Aarakocra's unarmed strike; rules do just what they say they do, and the Aarakocra's rules say that its unarmed strike deals exactly 1d4 damage, no Str mod.