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Побежать is the perfective form of the verb бегать “to run.” It incorporates one of the four most irregular verb stems in the Russian language:
|Present||No such thing as
Побежать literally means “to run”:
|Ваня побежал по дороге.||Ivan ran down the road.|
Although the verb does mean “to run,” it's actually used in conversation more often to mean “to take a quick trip” or “to be moving quickly” instead of literally running. The same is true for the English verb “to run” as well, of course.
|— Где мама?
— Она побежала в магазин.
|“Where is Mom?”
“She ran to the store.”
On the colloquial level the verb can almost mean the equivalent of the English “I'm out of here/I'm gone”:
|— Серёжа, не уходи. Останься, пока не напишем доклад.
— Нет, я побежал. Моя девушка ждёт меня.
|“Sergei, don't leave. Stay until we finish writing the report.”
“No, I'm out of here. My girlfriend is waiting for me.”
|Я не могу здесь остаться. Я побежал.||I can't stay here any more. I'm gone.|
Don responds: Oy! That was an embarrasing cut and paste error. The entry has been fixed. Thanks for bringing it to my attention!
Don responds: I've never seen a formal explanation of it. In English, of course, we do something similar in the phrases “I'm gone” or “I'm out of here.” In those cases the speaker is mentally picturing his departure as already completed, even though it hasn't happened yet. I suspect the same thing is happening in the Russian equivalents; that is, the speaker is mentally picturing his departure as a done deal, and so he expresses it in the past perfective.
1. Verbal communication is mostly used to express a message to people, to make them think that what you say is true. Consider messages like "[Treat me like] I'm out of here" or "[Считайте, что] я побежал [отсюда]". When you say something like this, you want others to understand that you've made a decision about your nearest actions, and you should be treated like you're already doing it. The part in brackets just repeats the reasons why people talk. No wonder that it's omitted in many languages, not only Russian.
2. If you add a bit of questioning or incentive intonation to the phrases described above, you'll get exactly the message people express with "Let's go shopping!": "[Пусть все считают, что мы] пошли по магазинам!". The "Let" ("Пусть") part merely got transferred into the omitted part of the phrase; that's why the resultant "Пошли..." phrase seems weird.