The main cost control will be national exchanges where people can go shopping for insurance from any state - regardless of where they live. Insurance companies that enact unreasonable rate hikes may not be allowed to participate in these exchanges - so they will be limited to hard selling people in their own states.
The Affordable Care Act says that parents can keep their kids on their private insurance until the kids reach 26; it prevents insurance companies from dropping insured people for getting sick (after all they paid their premiums) prohibits insurance companies from capping coverage - so they can't stop paying just because you have a serious illness that requires costly treatment, it also requires that some preventative care be free of deductibles and co-payments.
Starting in 2014 the law prevents insurance companies from denying policies to people who have pre-existing conditions. Before the Affordable Care Act, it was almost impossible for people with pre-existing conditions to get any health insurance, regardless of how much they paid. The law sets up an external review process - so that if a private insurer denies a claim, the patient can get a neutral party to review their decision instead of being forced to appeal directly to the insurance company.
The new law also gives tax credits to small businesses that offer health insurance to their employees. I really support this law. Once you actually read it and see what it does - it makes life better for most Americans.
If you want to read more about the law - there is a summary on this link. The complete act is available online at the Government Printing Office at this link.