If you look at the image, plain common sense indicates that the arms move independently of the wings. The wings aren't even attached at the shoulders, but right behind the shoulders, further reinforcing this.
Also, the arms clearly end in hands that look like hands, and the legs clearly end in feet that look like talons. So from the image, common sense tend to indicate that they hold their weapons in their hands, and do their talons attacks with their feet.
Now, if you look at the flavor text, several things point to the fact that they should not have too much problem striking with held in hands weapons while also flying:
They hail from a world beyond — from the boundless vistas of the Elemental Plane of Air.
Making denizens of the Plane of Air have extra-sub-par flying capabilities would go a bit contrary to the overall general principle that "it comes from another plane = it's better".
Nowhere are the aarakocra more comfortable than in the sky.
If they could attack only when standing on the ground, not while they are flying, then that would kind of directly contradict that sentence.
In battle, they prove dynamic and acrobatic fliers, moving with remarkable speed and grace, diving to lash opponents with weapons or talons before turning and flying away.
This very clearly states that they can attack with weapons while flying. Diving is still a form of flight, it is not like "falling".
Finally, look at the Aarakocra traits:
They gain Flight 50 feet, and that trait specifies that they can't be wearing medium or heavy armor. Nothing is said about a limitation on weapons, so there is no such limitation. Just like a Wizard casting Fly, is able to fight with weapons while flying, so too it is the exact same thing here.
In fact, apart from the obvious stat increase, Flight is really their only trait worth anything at all. And on land they move slower than most normal races! As for the Talons trait, that merely gives them competence with Unarmed Strikes, for a measly d4. But Talons doesn't give them a bonus attack at all. Almost every other form of attack will deal more damage. It's like a sickle that cannot be Disarmed, nothing more.
Also, if using overall D&D 5e game design logic: Equivalent features are not epic for some PCs, yet extra-gritty sauce for others. Aarakocras natural flight already pales in performance comparison to the Fly spell (slower, armor-type limited, self-only), but is overall better than Fly because it does not need spell slots or Concentration. But forcing Aarakocras to have to choose between flying or striking with weapons, that would seem quite a too steep penalty and make them very unfairly balanced. Or even comparing an Aarakocra, a natural born flier, to a paladin riding a pegasus: the paladin can strike while controlling his mount, so obviously the Aaracokra should be able to do the same too.
Forget previous D&D Editions. These new Aarakocras are just better. Also, much more fun to play that way as a player. More heroic, more adventurer-like, and less "beast-like".
However, unlimited flight, especially at low levels, can break a lot of encounters. Outside, just pepper your foes from way above with arrows. Inside, you can bypass 99% of traps (unless magic is so hyper-abundant that even at level 1 most traps are magical). But really this race has one thing and one thing only, so it should be at least a quite good ability.