The key to the question is the meaning of refer to.
We often just think this means linked to or corresponds with and people often use it like this.
So logically, such as connects to both public transportation and train or bus.
There is, therefore, no clear "correct" answer.
The most careful and thoughtful answer is train or bus because the definition of refer to we'd expect to be operative here
is alludes to, and such as alludes to the examples: trains or buses.
Smartest action: Maybe don't correct your teacher!
ell.stackexchange.com/users/11885/Jim Reynolds
'Such as' can be used in two different ways: 'Quadratic equations such as 3x^2+2x+1=0 can be solved using the quadratic formula' means all quadratic equations can be solved using this method, and the sentence is only giving an example for illustration.
'People such as him are very evil' means people like him are very evil and not all people are evil.
You need to determine from the context which case it is.
Hence without further information, 'public transportation such as train or bus' could mean public transportation in general (first sense above) or specifically trains and buses only (second sense above).
ell.stackexchange.com/users/None/
Such as means for example in this instance.
The example sentence is not a sentence, since at a minimum, it is missing a verb.
Regardless, we can still examine the phrase with for example : Public transportation for example train or bus
So trains and buses are examples of public transportation.
As for what such as refers to, I believe it is unclear.
One could argue that such as refers to public transportation since the examples are about that.
But such as also indicates what the examples are, so it could refer to the examples as well.
ell.stackexchange.com/users/36187/Em.
"Such as" is a phrase meaning "for example" or "like".
Another way to write the sentence would be: "Public transportation, like trains or buses,"
ell.stackexchange.com/users/40927/BuffyOverflow
The phrase such as train or bus is, in effect, an adjectival phrase modifying public transportation .
The nouns train and bus are subordinate to such as .
The only way the question makes sense to me is as asking what these words modify.
I wouldn't use "refer to" in this sense (I'd probably use "apply to"), but it's not impossible; I can readily imagine such a question being asked about a Latin sentence in which (as often happens, especially for poetic effect) an adjective is separated from its noun (but linked to it by gender and case).
ell.stackexchange.com/users/26707/Anton Sherwood
In the absence of commas, the use "such of" would refer to both of the following items.
Adding commas, could change the meaning, e.g. "He usually arrives via some form of public transportation such as a train, or a taxi."
The meaning of the latter may be enhanced by repeating the preposition, e.g. "He usually arrives via some form of public transportation such as a train, or else via a taxi."
ell.stackexchange.com/users/419/supercat
Example Sentence:    Public transportation such as train or bus.   Question: What do the words "such as" refer to?  Do the words "such as" refer to "public transportation" or to "train and bus"?  I had this question in my exam.
What does “such as” refer to in "Public transportation such as train or bus"?
ell.stackexchange.com
<grammar><meaning>
ell.stackexchange.com/questions/102036
ell.stackexchange.com/users/40983/Sergey Kozak
Although it is unclear what such as refers to, logically it connects to both  public transportation  and  train or bus. It is not clear what "such as" refers to. Due to the absence of commas, it could refer to both terms.
It is not clear what "such as" refers to. Due to the absence of commas, "such as" could refer to both terms.