Normally, the phrase "The adventure begins" uses the third person version of the verb.
Why is it "Let the adventure begin"? It is because this is a special tense?
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Because the form is imperative + bare infinitive and not present simple. Same as in: “Let him go”. You wouldn’t say “Let him goes”. In both sentences, go and begin are bare infinitives, not 3rd person singular of present simple.
Furthermore the adventure is a subject in the example “The adventure begins”, but in the other one “Let the adventure begin” it is an object. This is easier to spot in the example I gave, where the object is him. The form in which a personal pronoun in the 3rd person singular masculine can be an object is “he”, so when you see the word him you know it can’t be the subject of a sentence.