The Carleson embedding theorem gives a criterium by which to decide when for a measure $\mu$ the operator that takes a function on the real line to its harmonic extension (by convolution with Poisson kernel) on the upper half-plane is bounded from $L^2(\mathbb{R})$ to $L^2(\mathbb{R}^2_+,\mu)$.
Why is this theorem so significant? Why is this question interesting? Why is it called "embedding" theorem?
Because it's used to prove other theorems. See Operators, Functions, and Systems: An Easy Reading (Volume 1) by Nikolski, page 130.
This is subjective. If you do not find a result interesting, then it's not interesting to you.
Because a bounded injective operator between spaces of functions is often called an embedding.