The OP doesnt mean how can he see what the milky way looks like. He wants to know how we (mankind) know what it looks like.

I think by mapping the stars we can see (distance, location, speed), we get a partial shape. We can then compare that known shape with other galaxies we can see, and extrapolate that we must be in the arm of a spiral galaxy. I would bet that our known data points and math are exact enough that we have a rather detailed and accurate idea of what our galaxy looks like.