Although two people have already answered this, to get my post count higher, I've decided to answer it as well with an example.
The Shadow Scout has 6+ Dodge (and incidentally Hull 5).
a. Say I shoot it with a weapon that has 4 attack dice.
b. I roll the following: 4, 6, 2, 5
c. My weapon has no traits so I need 5's or more to hit it...that makes 2 hits!
d. But wait, the shadow player has Dodge 6+, he gets to roll 1 die per hit to see if he dodges...he rolls: 5, 6
e. He rolled (one) which means he "dodged" one of the hits! You can see however that if he rolled no sixes then both would have hit him with damage applied normally.
Does that make sense?