When it says orthogonal projections it must mean they are orthogonal with each other otherwise if it meant they were orthogonal lin transformations then they would be invertible- the only invertible projection is the identity map and if they were all equal to that they couldnt possibly sum to give the identity.
Here is the corollary I can't see why it should follow
ok it's not really a corollary but it uses the previous theorem. I PARTICULARLY CANNOT UNDERSTAND THE FIRST LINE OF THE PROOF- HOW HAS IT BEEN DEDUCED THAT V IS THE DIRECT SUM OF ALL THOSE EIGENSPACES WITH ALL THOSE $\lambda_i$ DISTINCT?
Those $V_{\lambda_i}$ are the images of the $E_i$ in the theorem.
The way that the spectral theorem is usually stated is that for a normal, complex operator $T: V \to V$, there is an orthonormal basis of $V$ consisting of eigenvectors of $T$.
In this proof, $V_{\lambda_i}$ is the span of the orthonormal set of eigenvectors corresponding to $\lambda_i$.