Learn German: Aus Eile habe ich die Schlüssel vergessen.

You could say, Weil ich es eilig hatte, habe ich die Schlüssel vergessen. (Because I was in a hurry, I forgot the keys). In that version, es acts as a fixed dummy object. The sentence here uses the noun Eile (haste) instead, which is a more concise way to express the exact same idea.

Why do we use aus here instead of vor to mean "out of"?

We use aus for reasons or motives behind an action, even if the action (forgetting) was an unintentional mistake. vor is typically used for involuntary, uncontrollable physical reactions to emotions, like vor Angst zittern (to tremble out of fear). Here, aus Eile tells us the logical circumstance that caused the oversight.

Why is there no article after the preposition? Shouldn't it be aus der Eile?

Abstract nouns describing a state, emotion, or reason typically drop the article when used with prepositions like aus and vor. It is standard to just say aus Eile (out of haste), aus Liebe (out of love), or vor Wut (out of anger).

Why does habe come before ich in this sentence?