How does a spell like "burning hands" that specifies that it damages "each creature" work with swarms.  Does "each creature" imply that it would utterly devastate the swarm (regardless of the swarm's total hit points) because it affects each bug/whatever individually?

You would think, but the rules don't say it does. DM may decide otherwise.

In a case where you're targeting a swarm with an area of effect spell, I would envision that only part of the swarm is actually being hit by the spell (versus each individual organism making up the larger swarm). For example, the fanning sheet of flames released from a burning hands spell cuts through the swarm of insects but doesn't necessarily incinerate each individual insect. The HP loss equates to all the little buggers being roasted by your spell, but some avoid/survive the attack (if the swarm had any HP remaining).

Speaking more mechanically, while the spell says that it damages each "creature" within its area of effect, the swarm rules do not view the creatures within it as being individuals. The swarm IS the creature. Otherwise, we'd have to make dozens of rolls for attacks and saving throws every time a swarm was involved. 

A swarm is treated as a single creature and any special effects are covered by immunities, resistances, or vulnerabilities.

• In 3.5e, swarms were resistant or immune to physical damage, but took normal damage from other things.
• In 4e, swarms were resistant to effects that target a creature (melee and ranged), vulnerable to effects that target an area (close and area).
• In 5e, swarms are resistant to physical damage and take normal damage from other things.

I generally like the 4e version the best