The rationale behind the mandate
Health insurance, like other kinds of insurance, works by creating "risk pools," which are groups of policyholders. In a typical risk pool, everyone pays insurance premiums, but only some will file claims. If a health insurance risk pool is large enough and has enough healthy people paying premiums, then there will be enough money available to cover the costs of those who get sick.
The rationale behind the individual mandate is that if everyone is required to have insurance—especially healthy people—the risk pools will be broad enough to lower premiums for everyone, even those with expensive medical conditions.