In the proof of this theorem, Arnold says "The mapping is an inhomogeneous linear mapping (why?)". Why is it obvious that it is so? Why can we write $\phi(T) = \lambda\phi(0) + C$ where $\lambda$ is the multiplier of the homogeneous equation?
Why does the assertion of the theorem follow from the fact that $\lambda \ne 1 $? Does it imply both assertions that the solution is of period T and there is exactly one such solution? Why?
This follows from the solution formula for linear first order equations, $$ (e^{-F(x)}y(x))'=e^{-F(x)}[y'(x)-f(x)y(x)]=e^{-F(x)}g(x)\\ \implies e^{-F(x)}y(x)-e^{-F(0)}y(0)=\int_0^xe^{-F(s)}g(s)\,ds \\ \implies y(x)=e^{F(x)}\left(y(0)+\int_0^xe^{-F(s)}g(s)\,ds\right) $$ where $F(x)=\int_0^xf(t)\,dt$. As you see, the coefficient is $\lambda=e^{F(T)}$ with exactly the described properties.
Obviously, for a periodic solution you would need $y(T)=y(0)$, which allows to compute that value as $$ y(0)=\frac{\int_0^xe^{-F(s)}g(s)\,ds}{e^{-F(T)}-1} $$ as long as $F(T)\ne 0$.