How do I get my iPhone screen to be invisible to the person standing next to me?

You can buy privacy screens that refract light different at different angles, so unless the person is right in front of the device the screen may appear partially obscured.

TECHO Privacy Screen Protector for iPhone 8 7 6s 6 - Amazon.com

Look for a Screen Protector that makes off angle viewing difficult. There are options on Amazon.com that do that.

As an example on Amazon, these are available for iPhone X and I'm sure for whichever model you have as well:

https://www.amazon.com/s/ref=nb_sb_ss_c_1_15?url=search-alias%3Daps&field-keywor ds=privacy+iphone+x+screen+protector&spr…

Buy a privacy filter.