Learn Danish: Jeg synes at historie er spændende. - I think that history is exciting.

In Danish, synes is used when you are expressing a subjective opinion or a personal feeling, like finding a subject exciting. You use tror when you are guessing or expressing a belief about a fact, like expressing that you think it will rain tomorrow.

Why isn't it historien if we are talking about history in general?

Just like in English, academic subjects and abstract concepts like historie usually don't take the definite end article when you talk about them in a general sense. Also, keep in mind that historie can mean both "history" and "story" depending on the context.

Does the adjective spændende need to change its ending to match the noun?

No. The word spændende (exciting) is an adjective ending in -ende. Adjectives with this ending are indeclinable, meaning they never change to add a -t or -e, regardless of whether the noun is common, neuter, or plural.

Can I leave out the word at in this sentence?

Yes, just like the word "that" in English, you can often drop at in everyday speech. Saying Jeg synes historie er spændende is perfectly natural and very common in Danish.