It means to hold a certain point of view - it's used to identify yourself as part of a group that believes in a certain model or theory or like. You'd probably never "subscribe to the idea that movie X is better than movie Y" unless that is well debated point in a fandom but you might "subscribe to the Austrian school of economics"
I think it's just the nature of the verb. You can say "We take the newspaper" to mean a similar thing (although the context would have to determine that you mean a subscription rather than stealing one, and that might be clearer if you named a specific paper, e.g. "We take the New York Times") and in that case it's definitely not an indirect object but a direct object (see definition 1.7 here: http://en.wiktionary.org/wiki/take)
But "subscribe to" is idiomatic in English... Actually, I was always under the impression that English usually considers nouns indirect objects only when that "to" is missing. (because I was taught that major sentence elements won't be in prepositional phrases, and while "to" is also in the English infinitive where it isn't a prepositional phrase, "to" indicating receipt of action and direct object is somewhat plausible as an adverbial phrase) "I gave him your card" has an indirect object, but I would have doubted "I took a lasagna to her". The internet at large does not seem to agree with me on this.
So you can ignore most of that paragraph if you like--more relevant is that in English it seems you need a direct object first before there can be an indirect object. Of course in German some verbs require the dative for their objects regardless, like "Ich danke dir", but it looks like abonnieren isn't one of them.
Yes, in English it is an indirect object, but in german not.
Just as we should not "translate term by term" the sentences, but instead learn the words usage, verb transitivity depends on the language, too.
For example, I'm a native portuguese speaker (Br), so I wouldn't expect that to be dative since for us there is no "to" equivalent in the sentence. We say "assinar o jornal", where "o" is just the article.
Hope that helps!
Native English speaker here - that sentence just doesn't make sense in this context. Only saying that you "subscribe the newspaper" makes it sound like you're subscribing the newspaper to something, not you subscribing to it. Unfortunately, I can't explain the reason why you have to say "subscribe to," but maybe someone who can will see this?
I believe it's because the plural form of newspapers is Zeitungen. The exercise here is looking for the singular.