What is a pluripotent stem cell?
A pluripotent stem cell is a cell that can differentiate into any of the major tissue categories. Every animal exists as a collection of pluripotent stem cells at an early stage of development: if you are looking for an evolutionary purpose, that's where you should start, and the purpose is pretty obvious: you start as a single fertilized egg and end up as a complex collection of tissues, so at some point that one cell has to become a collection of tissues, and there is an intermediate step which is a pluripotent cell.
None of this has anything to do with induced pluripotency, however.
What makes a cell pluripotent?
The answer to this question isn't really all that unique to stem cells: all cell types are differentiated from others based on their expression of transcription factors. Transcription factors determine what proteins get produced (including other transcription factors), which in turn determines how a cell functions and what role it plays.
Pluripotent stem cells express pluripotent stem cell transcription factors. Yamanaka started with embryonic stem cells, and looked for transcription factors that were special to those cells.
What does it mean to make a pluripotent stem cell?
Given that the identity of a cell is based on the transcription factors it expresses, the way to make one cell type act like another cell type is to change those transcription factors. A cell is a pretty complicated environment: there are lots of different transcription factors and lots of other types of proteins, so it's possible that it wouldn't be feasible to reprogram a cell this way.
However, Yamanaka and colleagues tried to artificially make an adult cell produce those transcription factors observed in embryonic stem cells. Indeed, they found that if they used the right combination of transcription factors in the right environment, the cells started to behave like embryonic cells, and they could further make those cells differentiate into different tissue types. These transcription factors are the so-called Yamanaka factors.
An induced pluripotent stem cell is a cell that has been transformed to produce transcription factors that are found in embryonic cells. Transcription factors are what make different cell types different from each other, by controlling everything else that the cell produces. All of this machinery is ubiquitous in development, the only intervention that the humans are doing is to make a cell express genes it would normally only express if it was an embryonic cell.
All of this information is already present in the Wikipedia article you linked.
Takahashi K, Yamanaka S (2006). "Induction of pluripotent stem cells from mouse embryonic and adult fibroblast cultures by defined factors". Cell. 126 (4): 663–76.
Takahashi K, Tanabe K, Ohnuki M, Narita M, Ichisaka T, Tomoda K, Yamanaka S (2007). "Induction of pluripotent stem cells from adult human fibroblasts by defined factors". Cell. 131 (5): 861–72.
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