"Run to the door" simply describes physical movement, but "run for the door" has emotional implications. It implies a sense of urgency. When you "run for the door" you really want to get out of the room/building in a hurry. Maybe you're frightened or embarrassed, or you just remembered that you have to be someplace else.

In American English, the phrase "I ran for the door" is commonly used to express urgency or a quick movement toward the door. The preposition "for" here implies intention or purpose, suggesting that the person is heading toward the door, often with a sense of haste or to escape something. While "I ran to the door" is also grammatically correct and means moving toward the door, it tends to focus more on the destination (the door) without necessarily implying urgency or a goal beyond reaching the door. In contrast, "ran for the door" emphasizes the action's purpose, such as trying to get out quickly or escape. So, the difference is subtle: - "I ran for the door" implies urgency or intent (often escaping or hurrying). - "I ran to the door" focuses more on the act of reaching the door, without necessarily implying the same urgency.

It it because you want the door right away. You are less interested in the direction of motion. Similarly, someone might say "I want out!". Grammatically, that is terrible but in urgent situations you focus on the thing you need right away, not precision or grammar.

Why do American use the phrase: I ran for the door. Wouldn't it be: Run to the door? The phrase "I ran for the door" is a common expression in American English, and it reflects a natural way to describe a quick movement toward a destination, typically in response to urgency or an immediate need to leave. The word "for" in this context indicates the direction or target of the action, emphasizing the intention of reaching the door quickly. This usage is rooted in idiomatic English, where "run for" something means to move swiftly toward a goal or destination. It's a concise and familiar way to convey both the action (running) and the purpose (heading toward the door). Americans, like speakers of other English dialects, use this construction because it efficiently communicates the sense of urgency and motion.