"Interessen der DDR" is "(political, strategic...) interests of the GDR", i.e. things or circumstances that serve the benefit of the state or its government.
The sentence doesn't mean that the news itself are against the GDR's interest, but rather their transmission (Übermittlung). So the "crime" that these brave people were accused of was an action that wasn't real espionage, but that was regarded as something quite close to it.
EDIT: Loong pointed out correctly that the clause that "Der Spiegel" is referring to deals with non-secret news (or more generally: data / information). So it is not about espionage (in the normal sense). The data could be literally anything, even as simple things as your passport data. As can be seen from this book on the GDR's law system(1) the "criminal act" was constructed from "the other side" learning about these (non-secret!) data. It was a classic ambiguous clause ("Gummi-Paragraph"), because it was completely at the discretion of the authorities whether it was "detrimental to the interests of the GDR" when someone learned about some data.
1) Marxen, Klaus; Werle, Gerhard: Strafjustiz und DDR-Unrecht. Band 5: Rechtsbeugung, Teil 1