The prayer you're referring to is called Havinenu.
The Rambam in Hilchot Tefilla 2:2 states that one who is rushed, or unable to concentrate, should say Havinenu. First he says the ordinary first 3, then Havinenu instead of the middle 13, and then the last 3. He thus fulfills his obligation.
In halacha #4 there, he clarifies that this does not apply during the winter, because he needs to ask for rain; or after Shabbat/Yom Tov, because there is havdalah.
I'm not entirely sure of the origins of the condensed form, but my guess is that the Sages said "better that people should say a short version than not be able to pray at all", because many people weren't able to remember the whole Shmoneh Esrei (this was before siddurim).