It's as simple as scrolling down until you see the "Is your portrait in a museum?" box.
When I first opened the app, I didn't really know what to do or where to go. Since the Google Arts & Culture app isn't dedicated solely to comparing selfies with art, there's a bunch of other stuff to sift through to get to the new function.
I opened the menu on the left, which is full of stuff but doesn't have a "take selfie" option. I closed the menu, and returned to the main "feed" of the app — a scrollable main content area laid out vertically. After scrolling past a few different featured stories, the white box surrounded by portraits appeared. Aha!
To note: While people in other countries can download the Google Arts & Culture app, this feature is exclusive to the US for now.