I just opened an Acorns account in 2020. How do I prove the amounts I’ve invested? And how do I file those investments in my 2020 taxes?

You don't need to prove the amounts you invested.  There is nothing to include on your return until you take money out.

This is a broker's account  and you should get a tax reporting form by the end of February which will report any taxable  earnings    such as   interest, dividends, capital gains and sales you earned/made during the year.  Log into your account to see when they will send those forms.  

https://www.acorns.com/support/taxes/will-i-be-provided-with-the-proper-tax-forms/