This is actually two complete clauses joined together with punctuation. It's not ideal style, but it is fairly common in journalism.
"For a country that sees itself as egalitarian, this smacks of class privilege"
Core sentence: This smacks of class privilege. Meaning, "this has the appearance of class privilege"--of giving special access to public goods to one particular social class or subgroup of society (usually the wealthy).
Add the modifying prepositional phrase "For a country that sees itself as egalitarian". Prepositional phrases are often used at the beginning of a sentence to "set the stage" or describe the general circumstances, without it necessarily being obvious which word they modify. "For a country that sees itself as egalitarian" means that the appearance of special privileges for the wealthy is particularly shocking, because the country's citizens tend to think that they have a pretty equal society.
The second sentence explains the first:
those going to Gymnasium tend to be disproportionately well-off.
"Gymnasium" is a type of college preparatory school in Germany and Scandinavia. This is identifying the class privilege mentioned in the first part: children of the wealthy are more likely to get the education that prepares them for university.
Structurally, the two clauses are complete. Each one has a subject ("this" and "those") and a predicate (everything else). It's probably just confusing because the sentences are fused together with a dash rather than being clearly separated with a period.