galaxy - Milky way: How do we know its appearance?

Milky Way is a young galaxy, filled with neutral hydrogen along its arms.
This hydrogen can undergo state transition, a rare event, which is compensated by the huge amount of hydrogen itself, i.e., even if this transition is rare it is still strong because a large amount of atoms can exhibit it.
The basics of this transition is explained by quantum physics through state-change in the energy levels of the hydrogen atoms, and can be summarized in the following picture:

The electron in the atom transits from a level with parallel spin (with respect to the nuclear proton), to the lower state, where the electron spin is antiparallel.
When this happens, the atom emits a photon, known as the 21 cm radiation, or hydrogen line.
This line falls into the radio domain, which can be observed by radio telescopes.

Since this radiation comes from rotating arms, doppler shift of the line can trace the distance of the emitter region.

And voilà, you have your map.

EDIT: ops, I missed the previous links where the answer was already given.