so why didn't they just harvest organs from dead people?

"they could just take them when anyone would die"

Do you mean people who had been murdered or killed in accidents? Unless they'd consented to donating their organs in the first place, the relatives of those people might not have approved. It would have been a legal headache.

Since the donors in this film didn't officially qualify as "human" in the conventional sense, lacking the same rights as "normal" people, it was simpler to use the healthy organs harvested from a clone. After all, they don't have relatives to mourn them.

I can think of two basic reasons, one was mentioned by Greg-233 already.

First, by harvesting organs they can guarantee a better quality - from young, healthy people, just after reaching maturity and before they start to age. The way it is done now, it is a catch-as-catch-can system, so they only get what happens to be available. Statistics probably would demonstrate (I don't have that information) that most organs become available after the donors have reached at least middle age.

Second, in order to guarantee the population of "real people" access to donations when they need them, the state would have to ensure sufficient supply. Whereas, in our actual circumstances, there is never a sufficient supply of organs for donation (also perhaps due to the fact that some don't consent to it), in the "Never Let Me Go" world, the state can plan and create enough clones to guarantee there is sufficient supply for the needs of all "real people".