From one of the great legends of the American people:
In this situation the word order is setting emphasis. Because коробка is at the end of the sentence, the emphasis is that they're giving you specifically a box (just like you've always wanted).
If it had been at the beginning, the emphasis would have been that they're just giving you the box (how can they afford to just give away boxes?!)
Коробку*. The "direct object" (the thing that is in accusative - коробку here) is the thing being acted on directly by the verb. The verb here is give. The box is the thing being given. The "indirect object" (the thing in dative - мне) is someone who is receiving the result of the action, if that makes sense. Technically with the grammar you can switch мне и коробку and it will retain the same meaning. The sentence here though is emphasizing what is being given, not to whom, though.
Like in many languages, a single verb suffices for the present tense (and in many cases in the past and future as well). In this way, давать (imperfective) can be translated as both "is/are giving" or simply "give" when conjugated in the present tense.
Since Russian doesn't use articles, this can be either "the" or "a" box.
The audio isn't pronouncing каробку correctly. The middle "o" is stressed and should sound like an "o", not an "a". Here's some correct pronunciations: (something like Kah-ROHB-Koo) https://forvo.com/word/%D0%BA%D0%BE%D1%80%D0%BE%D0%B1%D0%BA%D1%83/#ru