The disk structure is even more easily seen in the infrared, where dust extinction is much less than in the optical. See for instance this image from the 2MASS infrared survey:
Stars are not really (as Milan van Dijck suggests) used for mapping the spiral arms. Rather the gas clouds of gas in between stars are used, more specifically neutral hydrogen observed through its 21 cm observation. The figure below (from Oort 1959) shows the resulting map. As you can see, its not really as well-defined as seen in many artist's-conception pictures.
Neutral hydrogen is fuel for stars. Massive stars ionize the gas around them, and since star formation mainly takes place in the spiral arms, the gas clouds are accompanied by pockets of ionized hydrogen, as seen in this map from Georgelin & Georgelin (1976):
The physical properties of the Milky Way (stellar population, gas fraction, color, kinematics, etc.) are also similar to other spirals.
It is clearly a disk galaxy since the Milky-Way is a stream across the sky and so we conclude that the Galaxy is a highly flattened structure.
We observe that external galaxies which are highly flattened are almost all spirals of some form. (Also the stellar populations and arrangement of globular clusters are similar to those of external spirals)
You can see that the milky way appears as a stream across the night sky. We can calculate the distance between us and the stars in this stream, therefor we also know the distance between those stars (by using the Pythagorean algorithm for example). Because we now know the distance between them, we know their relative position of each other (by triangulation).
If you would then construct a 3D model by using these relative positions it would look like a spiral.