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The downshift causes the engine to turn at a slower rate. This causes the wheels of the car and the engine to be more similar and gives more of an emphasis on producing power versus a top speed which is achieved in a higher gear.

Alternate answer:

An internal combustion engine only develops power when it is turning between a narrow band of revs. usually between 3000 and 6000 rpm on the average car.

The hill puts a heavy load on the engine and causes it to slow down. Shifting to a lower gear (actually a higher gear ratio), allows the engine to speed up and develop more power to cope with the hill.

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