Peter is represented by the string quartet because he is the main character.
In most types of orchestral work that is NOT a concerto (that is, an extended solo performance of an instrument), the string section is the "main character", so to speak. They are the largest section by far in the orchestra, and often employ the "body" of many works, with brass and woodwind being the added "color" or "sound" in a symphonic work.
This runs true of virtually any "full orchestral work" where strings are present. Sometimes the brass may get a solo here and there, and the woodwinds may carry the melody, but the main body of the piece is is the string, though they may sometimes take a backseat and give the spotlight to some other instruments.
Of course, "Peter and the Wolf" isn't a completely full orchestral piece, but it uses a good majority of the instrumentation employed in full symphonic composition. Peter, as the main character, leads the story, and everything revolves around Peter (more or less), like how orchestral music is often centered on the strings.
This also makes sense in the explanation, as it states,
"Thereby, the children learn to distinguish the sonorities of the instruments during the performance of this tale.
Part of the reason for the existence of "Peter and the Wolf" is to teach children about the different sounds of instruments and how they all work together to make music (or tell a story). By having the entire string section (not merely just say the violins) represent Peter is important, because it engrains into the children's early about the importance of the strings in an orchestral piece.
The instrumentation also creates auditorial imagery. By listening to the leitmotif of Peter, one can imagine a jolly little lad skipping around, full and vibrant with energy.
The final reason is I think commented a bit by Peter Smith, and it's to distinguish between texture and timbre. It's very interesting to note that each instrument has its own unique timbre:
- Flute: Light, airy, and clear.
- Oboe: Darker, more "reedy", (kind of sounds like a duck, doesn't it?)
- Clarinet: Full, airy, and "fluffy"
- Bassoon: Darker and heavier than the oboe.
- Strings: Light, majestic, and vibrant
-French Horns/Brass: Menacing and loud
- Timpani: Short and loud
In a full orchestra, strings tend to function as one unit, rather than detached. This works well in their favor because strings have a timbre that harmonizes well together. So I think having the full string section represent Peter probably speaks to the complexity of humans, similar to how the hunters are represented by a mixture of trumpets and woodwinds.