I will try to be short so as to not make the academic idea of "sonata form" sound even more complicated than many academians have already made it.
At its base, sonata form is structure of the highest order. It dictates exactly how a piece will play out. It makes music rather predictable, and it provides an even platform for listening. This is no different than popular music of today.
Nearly everybody does Introduction (usually instrumental)-Verse-Chorus-Verse-Chorus-Bridge-Chorus. This is no different.
Sonata form is: Introduction, Exposition, Development, Recapitulation, Coda.
It speaks of the Classical Period in that it speaks of structure--a structure that is copied even today. Because you have two men (Haydn and Mozart) who almost exclusively represent the Classical Period, and these men favored this form, it represents what most people know of the period.
As Beethoven bridges the gap between the Classical and Romantic Periods (he is the last Classical and first Romantic composer) his desire to move away from the "stodgy" sonata form solidifies its place IN the Classical Period while other forms live outside the Classical period. It lived and died there.
That's the simple version--I hope it helps.