You just check the box in the tax form. If audited, you will need proof so that the IRS does not deny the exception. You should save any documents related to the buying and selling of a home for as long as you own the home plus 7 years after you sell.
For a Roth IRA you can withdraw the original contributions (that you already paid tax on) at any time for any reason and owe no tax.
To withdraw your gains (income), then:
You can use the first time home buyer exception to withdraw up to $10,000 of earnings without penalty. If the Roth IRA has been opened at least 5 years, the withdrawal is also exempt from regular income tax. Withdrawing more than $10,000 of earnings will result in regular income tax plus a 10% penalty, unless you are over age 59-1/2.
