As a reflection of the importance the Graduate School places on the ability of its students to communicate effectively, the Graduate School requires all new students whose native language is not English to have their English evaluated.
I can see how you would have some trouble with that sentence. I don't think it is very well written. as a reflection of is not particularly well suited as a modifier of the verb requires.
If it had said "has instituted a requirement" then as a reflection of would work better.
Let's reorder the clauses and evaluate:
The Graduate School requires all new students whose native language is not English to have their English evaluated, as a reflection of the importance the Graduate School places on the ability of its students to communicate effectively.
Not really any better. That sentence is even a little more jarring. Let's try another reordering:
The Graduate School requires, as a reflection of the importance the Graduate School places on the ability of its students to communicate effectively, all new students whose native language is not English to have their English evaluated.
A little clearer (if clunky). We could use "it" instead of repeating "Graduate School" and use a that-clause:
The Graduate School requires, as a reflection of the importance it places on the ability of its students to communicate effectively, that all new students whose native language is not English have their English evaluated.
or
As a reflection of the importance it places on the ability of its students to communicate effectively, the Graduate School requires all new students whose native language is not English to have their English evaluated.
Now let's really simplify:
The Graduate School places great importance on the ability of its students to communicate effectively and therefore requires all new students whose native language is not English to have their English evaluated.