The purpose of the 5-year rule on Roth contributions is relatively straightforward - to require that tax-free growth for retirement purposes be done for the long-term, which means the account must be maintained for at least 5 years (in addition to meeting one of the other requirements).
To understand the purpose of the 5-year rule for Roth conversions, an example of what would occur if it wasn't there might help.
Joe is 40 years old and would like to tap the $50,000 of funds in his traditional IRA. If he takes a withdrawal now, he will be subject to ordinary income, and a 10% early withdrawal penalty. If Joe converts his IRA to a Roth IRA, he will also be required to report the amount as ordinary income; however, he can now take a withdrawal of his "after-tax" principal from his Roth IRA (the conversion amount) without an early withdrawal penalty. The end result: Joe could entirely circumvent the IRA early withdrawal penalty by simply doing a Roth conversion first and taking the money thereafter, so the 5-year conversion rule is designed to prevent this!
Continuing the prior example, if Joe completes his conversion and waits 5 years, he will be eligible to withdraw his Roth conversion principal without any early withdrawal penalty; this is true because he has met the 5-year requirement for conversions, even though he would only be age 45 at the time. Thus, while the 5-year conversion rule prevents individuals from outright dodging the early withdrawal penalty from their IRAs, it does allow them to potentially gain access to their IRAs prior to age 59 1/2, albeit with a 5-year waiting period! (Though notably, any gains on Joe's conversion would still be taxable, as even if he has met the 5-year requirement for conversions and contributions, he has not met the 59 1/2, deceased, or disabled requirement to receive tax-free qualified distributions from his Roth IRA.)
Accordingly, it's also worth noting that because the 5-year rule for Roth conversions merely leaves the withdrawal of conversion principal potentially subject to the early withdrawal penalty, any other exceptions to the early withdrawal penalty can still shelter the Roth conversion amount from the penalty. Thus, withdrawals within 5 years of conversion by someone who is already over age 59 1/2 are not subject to the early withdrawal penalty, regardless of the 5-year conversion rule, simply because being over age 59 1/2 itself is an exception to the penalty!