Withholding taxes when removing an excess contribution?

When you withdraw an excess contribution to an IRA, you are also expected to withdraw any gains attributable to that contribution, and those gains are taxable income (not classified as capital gains) to the IRA holder. For example, your IRA was worth $40K at the end of 2018 and you contributed $10,000 to the IRA for 2019, thus making a $4000 excess contribution. Because you have been fortunate in your IRA investments, your IRA account is worth $60K when you go to withdraw your excess contribution. If you take out only the excess $4000, you have made a tax-deferred profit inside your IRA from that excess contribution which profit you wouldn't have made at all if you hadn't made the excess contribution. That's why you must withdraw the excess gains from that excess contribution, not just the excess contribution. You are liable for tax on that excess gain, and Vanguard just wants to know if you want to have some tax withheld right then and there. If you select 0% (a.k.a. no withholding), you will pay the tax due when you file your return. Tax returns are where you get to compute the tax due and then tell the IRS what you think is due, how much was withheld already, and send a payment for any shortfall or request a refund of the excess withholding. Thus, selecting 0% is not harmful except when you expect you might be penalized for not having paid sufficient tax on a quarterly basis.

I am under the impression that taxes will be withheld when I fill out the 2019 taxes so I am confused as why I would do it here? What would happen if I don't withhold taxes?

When you are completing your return taxes aren't withheld, the return is where you document everything and settle the account.

If the calendar year hasn't ended then is can make sense to have the investment company withhold money and send it to the IRS and state tax authority. This is especially true if your tax situation is such that you will ow a lot of money when you file your return. The federal government, and maybe the state government, can impose a penalty for having too little withheld.

If the investor is in that situation they might want to have taxes withheld to make sure they don't get hit with a penalty. You will have to tell them how much to withhold because every tax situation is different.