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Cold air holds less water vapor (lower humidity). As the warm, most air inside changes temperature inside your walls to the outdoor temperature, the excess moisture that the colder temperature cannot hold condenses inside your walls. You need a vapor barrier on the warm-in-winter side to prevent that from happening. The vapor barrier traps the higher-humidity in the warm air inside the room. The temperature "flows" through the barrier, but the water vapor cannot, so it can't thereafter condense inside your walls.
It is also possible to remove the mildew with bleach treatments before you spray the insulation in, but the most sure-fire way to eliminate the mold that might be in your walls is to rip out the walls, treat the studs, and install insulation with a proper vapor barrier.
See the other thread on mold for other things you can do to stop it.