Why does $\displaystyle f'(x)=0 \; \forall \; x\in\mathbb{R}\implies f(x)=c $?
Well, you have an infinite family of solutions depending on your initial condition. However, the fact that the RHS of your DE, namely 0, is Lipschitz and continuous, is enough to guarantee the uniqueness of the solution to an initial value problem (an initial value problem is your original DE plus an initial condition to determine the constant of integration). Does that make sense?
Because of the Picard-Lindelof theorem. If you're curious, by all means investigate the proof of that theorem.
For this specific question, you don't need the full "Picard-Lindelof", just the mean value theorem.
Let $\displaystyle x_0$ and $\displaystyle x_1$ be two distinct values of x: $\displaystyle x_0\ne x_1$.
By the mean value theorem, $\displaystyle \frac{f(x_1)- f(x_0)}{x_1- x_0}= f'(c)$ where c is some number between $\displaystyle x_0$ and $\displaystyle x_1$. If f'(x)= 0 for all x, then f'(x)= 0 so $\displaystyle f(x_1)- f(x_0)= 0$ and $\displaystyle f(x_1)= f(x_0)$.