I've read Robert Gordon's article titled 'Folk Psychology as Simulation'. This is a classic article that represents the so-called 'simulation theory' about understanding the other people's mind.
The central criticism of this article is that our capability of understanding the other people's mind and behavior is not based on theoretical inferences as was thought by the theory of mind approach (the "theory theory of mind") ;if A is in states S1, S2, S3, etc., and conditions C1, C2, C3 obtain, then A will (or probably) do X. For example, If A feels thirsty and he sees a bottle of water on the table, then he will drink the water. Gordon writes that our understanding of the others is not 'nomological reasoning' like this example, but it is 'practical reasoning'.
What he calls 'practical reasoning' is a kind of simulation (the "simulation theory of mind"). Using the example of chess players, he explains;
[T]he task is to answer the question, 'what would I do in that person's situation?' For example, chess players report that, playing against a human opponent or even against a computer, they visualize the board from the other side, taking the opposing pieces for their own and vice versa. Further, they pretend that their reasons for action have shifted accordingly: whereas previously the fact that a move would make White's Queen vulnerable would constitute a reason for making the move, it now becomes a reason against; and so on. Thus transported in imagination, they 'make up their mind what to do.' That, they conclude, is what I would do (have done). They are 'putting themselves in the other's shoes' in one sense of that expression: that is, they project themselves into the other's situation, but without any attempt to project themselves into, as we say, the other's 'mind'.
Gordon, R. M. (1986/1995). Folk Psychology as Simulation. in M. Davies and T. Stone (Eds.), Folk Psychology: The Theory of Mind Debate. Oxford: Blackwell. p.63
Thus, according to Gordon, we 'put ourselves in the other's shoes' in order to understand the other person. We project ourselves into the other's situation, we perceive things (virtually) in that person's perspective, and we feel and think as if we were that person.
In other words, what Gorson states is that our understanding of the others is not a kind of theoretical inference from third person's (objective) perspective, but is practical simulation from virtual first person's (subjective) perspective. And thus we can understand and predict other people's behaviors. From phenomenological point of view, here we could find something common with Husserl's notion of 'Einfühlung'.