If a Monk fights only with their fists at level 1 and has only one attack during their Attack action, do they do 2d4 damage because they're using both hands? Or is it only one hand per attack? Meaning he would need to be level 5 where he gets the Extra Attack feature to do the same thing?
One unarmed strike does 1d4 (+ attribute modifier) damage for a first level monk.
An unarmed strike doesn't necessarily have to be a hand/fist attack. It could also be a head-butt, a kick or something like that.
The monk could use its bonus action to do an additional unarmed strike for another 1d4 (+attribute modifier) with its Martial Arts ability. Each attack requires a separate attack roll.
They do 1d4 damage plus their ability modifier
The text states:
You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon.
An unarmed strike can be narratively described as a single punch, multiple punches, kicks, etc.
Mechanically, however, it is the same regardless of how you are describing it. An unarmed attack is an attack made with a non-weapon.
1d4 + (Str or Dex)
A 1st level Monk may only make 1 attack as part of their Attack action. Their unarmed strike will do 1d4 damage (as per Martial Arts) and they can choose to use either their Strength or Dexterity modifier as a bonus to the attack and damage rolls. This attack does depend on any held items or free hands. It can be made with any part of the body: foot, knee, elbow, fist, head, etc.
When a Monk makes an attack with their unarmed strike or with a monk weapon as part of their Attack action, they are eligible to spend their bonus action to make an additional unarmed strike (as a new attack, roll again).
At level 5, the Monk will be able to make 2 attacks every time they take the Attack action. At that point their Martial Arts would then be a 1d6, but they would still roll for each attack separately.
You can deal more than 1d4 damage, but it's not about number of hands
If a Monk fights only with their fists at level 1 and has only one attack during their Attack action, do they do 2d4 damage because they're using both hands? Or is it only one hand per attack?
No, a Monk doesn't do 2d4 damage because of they're using both hands. And no, there is no "one hand per attack" rule for unarmed strikes.
Unarmed Strike is something more complicated than just standing and punching left, right, left, right. See Basic Rules, Making an Attack:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow
Starting from the 1st level, a Monk has their Martial Arts class feature. That feature, not number of hands the Monk has, allows him/her strike more than once per turn, using their bonus action:
When you use the Attack action with an unarmed strike or a monk weapon on your turn you can make one unarmed strike as a bonus action.
The Monk can make this extra strike even when both his/her hands are occupied. See Could a Monk holding two weapons still allow for the bonus Unarmed action?
So yes, you can deal more than 1d4 damage. Moreover, starting from 2nd level you can spend your Ki points to strike even more:
Flurry of Blows
Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.
A 1-level Monk can strike twice in one 6-seconds round, and a 2-level Monk can strike thrice. Lore-wise, it is because Monk is fast, agile and skillful, not because she/he has three hands.