I am reading a source that says:
Let $L \to M$ be complex line bundle. Define $\Omega^k(M,L)=C^\infty(M,L \otimes\bigwedge^k T^*M)$.
Can someone explain thoroughly what this means? What does it mean for k-form to have values in $L$? Can someone explain what the tensor product here means?
A vector bundle (up to isomorphism) is determined by its transition maps. Let $E$ be a vector bundle of dimension $n$. Let $(U_\alpha, \varphi_\alpha)$ and $(U_\beta, \varphi_\beta)$ be two trivialization with $U_\alpha \cap U_\beta \neq \emptyset$. This means $$\varphi_\alpha : E\big|_{U_\alpha} \longrightarrow U_\alpha \times \mathbb{R}^n$$ is a diffeomorphism and it is the same for the $\beta$ pair. The definition of vector bundle requires that $$ \varphi_\beta \circ \varphi_\alpha^{-1} \big|_{U_\alpha \cap U_\beta}: U_\alpha \cap U_\beta \times \mathbb{R}^n \longrightarrow U_\alpha \cap U_\beta \times \mathbb{R}^n$$
is given by the form $(x,v) \mapsto (x, g_{\beta \alpha}(x) v)$ where $g_{\beta \alpha}(x)$ is smooth matrix-valued function and is called a transition map. Conversely, if you have a open cover $\{ U_\alpha \}$ and transition map $g_{\beta\alpha}$ on each ${ U_\alpha \cap U_\beta}$. You can glue the trivial bundle $U_\alpha \times \mathbb{R}^n$ together in an obvious way. The resulting topological space is a vector bundle diffeomorphic to $E$.
So given two vector bundle $E$ and $F$ with transition map $g_{\alpha\beta}$ and $h_{\alpha\beta}$. The tensor product $E \otimes F$ is the vector bundle associated with the transition map $g_{\alpha\beta} \otimes h_{\alpha\beta}$. (Recall if you have two linear transformation $T : V \longrightarrow V$ and $S : W \longrightarrow W$, $T \otimes S$ is the map $v\otimes w \mapsto T(v)\otimes S(w)$. )
In your case, take $E = L$ and $F = \bigwedge^k T^\ast M$. The elements of $\Omega^k(M,L)$ are the sections of this vector bundle. Locally, they really can be write as finite sums of $L$-coefficient $k$-forms.