Suppose I have a set of smooth $k$ functions $F_{i}: M \mapsto \mathbb{R}$, where $i=1,...,k$ on a smooth manifold $M$. I encounter a theorem that states "Assume $dF_{i}$ are independent at each cotangent space $T_{x}M$".
What does this mean if I take $k=1$, i.e. I have only one $F$? How does then the independence condition look? Do I simply need that $dF$ is not identically vanishing, or something more?
I think you mean $F_i :M \rightarrow \mathbb{R}$ so I will go with that assumption.
The derivative of each of these maps is $dF_i :TM \rightarrow \mathbb{R}$. We define $$TM := \{(x, v(x)) : x \in M, ~ v(x) \in T_x M \}$$ and so if we restrict to $T_x M$ we are effectively looking at the maps $dF_i(x) : T_x M \rightarrow \mathbb{R}$. By how differentiation on a manifold works these maps will be linear so the question is asking you to assume that $dF_1(x), \ldots, dF_k(x)$ are linearly independent as linear maps.