How to prove the given sets are equal? 
I did not understand the highlighted part(in red). 


*

*How is the equality obtained? One inclusion is clear namely, the left hand side is a subset of the right hand side.

*Why is this an open subset of R^n?
 A: Two set-theoretic facts to use here:


*

*The preimage of an intersection is the intersection of the preimages.

*If some subset $B$ lies in the image of a map $f$, then $f(f^{-1}(B))=B$.


Applying these to our bijection $\phi: U \to \phi(U)$, we get
$$\phi^{-1}\big(\phi(U) \cap (\mathbb{R}^n \times \{0\})\big)\overset{(1)}{=}\phi^{-1}(\phi(U)) \cap \phi^{-1}(\mathbb{R}^n \times \{0\})=U \cap (U \cap N)=U \cap N.$$
Working from right to left, hit the whole thing with $\phi$ again:
$$\phi(U \cap N)=\phi\big(\phi^{-1}\big(\phi(U) \cap (\mathbb{R}^n \times \{0\})\big)\big)\overset{(2)}{=} \phi(U) \cap (\mathbb{R}^n \times \{0\}).$$
As for your second question, recall that charts are defined to be homeomorphisms from open subsets of your manifold to open subsets of a Euclidean space. So $\phi(U)$ is open in $\mathbb{R}^n \times \mathbb{R}^{m-n}$, hence the intersection $\phi(U) \cap (\mathbb{R}^n \times \{0\})$ is open in $\mathbb{R}^n \times \{0\}$ by the definition of the subspace topology. 
