Struggling to understand wavevector K for deriving energy levels in reciprocal space
The Hamiltonian for a particle in a periodic array of atoms is invariant (doesn't change) under translation by any vector of the form $n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3\vec{a}_3$, where the vectors $\vec{a}_1,\vec{a}_2,\vec{a}_3$ for the unit cell and $n_1,n_2,n_3$ are any integers. The set of all such translations form a group. 

When a Hamiltonian is invariant under all elements of some group, it follows that its eigenfunctions can be written as functions of definite symmetry with respect to that group. A function of definite symmetry with respect to the group described above turns out to be characterized by a "wavevector" $\vec{k}$. This is completely analogous to situation for the hydrogen atom, where spherical symmetry implies $[L^2,H]=0$ which in turn implies that total angular momentum is conserved and that our eigenfunctions possess a quantum number $l$ (along with $n$ and $m$) that quantifies the wavefunction's total angular momentum. To make the analogy clear:


Hamiltonian of hydrogen atom commutes with arbitrary rotations $\to$ eigenfunctions of Hamiltonian have quantum number $l$
Hamiltonian of particle in periodic potential commutes with arbitrary translations $n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3\vec{a}_3$ $\to$ eigenfunctions of Hamiltonian have wavevector $\vec{k}$.


A function $\phi_{\vec{k}}$ of wavevector $\vec{k}$ happens to have a special behavior under group operations:
$$
\phi_{\vec{k}}(\vec{x} + n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3\vec{a}_3) = e^{i\vec{k}\cdot(n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3\vec{a}_3)} \phi_{\vec{k}}(\vec{x})
$$
At this point we usually note that if two wavevectors $\vec{k}$ and $\vec{k}'$ differ by some $m_1 \vec{b_1} + m_2 \vec{b_2} + m_1 \vec{b_2}$, where 
$$
\vec{b}_i = 2 \pi \frac{ \sum_{j,k=1}^3 \epsilon_{ijk} \vec{a}_j \times \vec{a}_k}{\sum_{m,n=1}^3 \epsilon_{imn} \vec{a}_i \cdot \left( \vec{a}_m \times \vec{a}_n \right)}
$$
and $m_1,m_2,m_3$ are integers, then
$$
\left( \vec{k'} - \vec{k} \right) \cdot(n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3\vec{a}_3) = 2 \pi \left( n_1 m_1 + n_2 m_2 + n_3 m_3 \right)
$$
so that the wave vectors have equivalent symmetry properties. This allows us to restrict our consideration of wavevectors to those of the form $c_1 \vec{b}_1 + c_2 \vec{b}_2 + c_3 \vec{b}_3$, where $ 0 \leq c_j \lt 1 $. More commonly we equivalently restrict the possible wavevectors to lie in the Brillouin zone.

As to why the concept is introduced, the answer is that recognizing that our energy eigenfunctions can be classified by some symmetry often enormously simplifies problem solving. Using again the hydrogen atom as an example, recognizing the Hamiltonian's spherical symmetry allowed us to break up a 3-dimensional Schrodinger equation into three 1-dimensional differential equations, which is an enormous simplification. For a periodic pontential, knowledge that our eigenfunctions have a wavevector $\vec{k}$ allows us to find the eigenfunctions by solving a differential equation over a single unit cell (as opposed to all space).