Explain how doppler shift supports the theory of an expanding universe.

You've got things in scrambled order.
Real science doesn't use observed evidence to support theories.
Real science creates theories to explain observed evidence, and then tries
as hard as it can to prove that the theories are wrong, by testing them.

#1).  We observe that light from distant stars and galaxies
is shifted toward longer wavelengths than it had when it
left the source.

#2).  We observe that the farther from us the stars and galaxies are,
the MORE their light is shifted from the wavelengths it had when it
left the source, toward longer wavelengths.

#3).  As of now, the Doppler shift which relates change in wavelengths
to radial motion between the source and the observer, is the ONLY
thing we know that can change wavelengths like what we observe.

#4).  If it IS Doppler shifting that's causing what we observe, then
all of the distant stars and galaxies must be moving away from us.
AND, the farther one of them already is, the faster it's moving away.

HOW CAN WE EXPLAIN ALL OF WHAT WE SEE ?

Since the Doppler mechanism is the only way we know of that wavelengths
can get shifted like what we see in every direction from us, our attempt to
explain all the wavelength shifts we see leads us to formulate the theory
of the expanding universe.  And THAT also leads to the "big bang" theory.

Note #1:
If we ever discover ANOTHER way that wavelengths can get shifted,
it could dump the whole expanding universe theory out the window, AND
the "big bang" theory along with it.   

Note #2:
The whole Doppler shift observation means we have to measure the
wavelengths of the light we see here on Earth, and then write down
how far it shifted from whatever the wavelength was when it left the
distant star or galaxy.
HOW DO WE KNOW what it was when it left the distant star or galaxy ? ? ?
To me, THIS trick is even more awesome than the theories we've built
from it !   

The correct answer is waves