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The reason that Moses was told to effect a cure is that people had already been bitten, and simply sending the snakes away would have been insufficient to stem the rate at which people were dying. But if you look closely at the language in 21:7, what the people were really asking Moses for was forgiveness (cf: Rashi and Or haChayim on that verse). As Or haChayim points out, forgiveness has to come before the effects of that forgiveness can come, which is why Moses needs to build a serpent for people to look at before the rest of the snakes can be taken away.
As for how the bronze serpent provided a cure, you might be interested to look at Ramban, whose interpretation is reminiscent of 19th century homeopathy (!), or consider the following passage from the Mishna: