This spell is just super awkwardly worded
No matter how you interpret the spell, you're going to run into problems.
If it's flavor text, there's nothing that sets it apart as such. And in other spells (Call Lightning, for example) the "flavor text" and what is clearly rules text intertwine within the same sentence.
On the other hand, if it's not flavor text, it's worded as part of the spell's effect, rather than a requirement. We have examples of requirements elsewhere, and they're pretty explicit: "The spell fails if..." in Call Lightning, for example.
Or in Speak with Dead: "The corpse must have a mouth, and can't be undead."
So, you can cast Burning Hands with just one hand free. One free hand satisfies the Somatic component, and there isn't a restriction in the spell text.
And that's a problem, because the effects of a spell happen, even if physics suggests otherwise.
So, when a one-handed man casts this spell, a few things happen:
What does that look like? Does a spectral limb materialize? Does their severed hand fly back to them from the grave? Do they just hold their stump in roughly the right location?
You really have a few options here, and barring a developer stating which was intended or an "official" ruling from organized play, one's as good as another. This is a case where unclear writing moves things into the DM's court.
It's clearly a requirement of casting the spell
...Don't read too much into how it's worded.
It's clearly flavor text
...Those don't read like the rules, therefore they aren't.
It's clearly rules text, but with a lot of wacky edge cases
...Like having one hand bound, or not having a hand, or not having catalytic agents to create a fire (maybe not that last one).
The spell fails when it attempts to do the impossible
...If any part of the spell's effect fails, the entire spell fails.
The spell does what it can, and the rest doesn't happen
...You don't touch your fingers, because you can't. But you still get fire because you can(?).
Occam's Razor was brought up in the comments, and this is indeed the tool you need to apply to this situation. The problem is that there are several "simplest" solutions to this situation.
For the sake of avoiding humorous situations, let's consider a situation likely to occur in an actual game: A humanoid caster wielding a one-handed sword in his hand.
Outcome: The caster cannot cast the spell.
Because: The caster is unable to touch both thumbs together and spread his fingers.
Assumption: If part of a spell fails, the entire spell fails.
Outcome: The caster touches two thumbs together and spreads fingers, around the grip of his sword.
Because: It is physically possible for the caster to hold a weapon while making the proscribed gesture.
Assumption: The spell is completed in the most plausible way that agrees with the rules.
Outcome: The caster holds out one hand with fingers spread, while holding his sword in the other. Flames shoot from the outstretched hand.
Because: The caster performs as much of the spell as possible.
Assumption: Spells complete to the greatest extent possible.
This gives us three assumptions, all unsupported by the rules. Which is the greater assumption?
If part of a spell fails, the entire spell fails.
The spell is completed in the most plausible way that agrees with the rules.
Spells complete to the greatest extent possible.