Suppose preferences are strictly monotone and strictly convex and consider any point \((x_{1},x_{2},x_{3})\). Geometrically, consider the plane (in green in the diagram above) going through the point \((x_{1},x_{2},x_{3})\) that is tangent to the indifference surface (in blue) going through the point \((x_{1},x_{2},x_{3})\). The plane corresponds to some combination of prices and income \((p_{1},p_{2},p_{3},y)\). If we put the consumer in a situation where he has disposable money \(y\) and is faced with prices \((p_{1},p_{2},p_{3})\), he will choose the point \((x_{1},x_{2},x_{3})\). From that we will be able to learn that the indifference surface going through the point \((x_{1},x_{2},x_{3})\) is tangent to the plane corresponding to \((p_{1},p_{2},p_{3},y)\). Intuitively, since we can do this for all points, we can recover the indifference surfaces. Using strict monotonicity, this allows us to then infer the preferences, because it will allow us to rank all the indifference surfaces.
In fact, the procedure works in any dimensions.