The message means that (for whatever reason) the checksums used to verify the source files (see Integrity on the Arch wiki) in the PKGBUILD are not matching the checksums produced by running
sha512sum on the file makepkg downloaded. This could mean that someone has managed to make your computer download a different file than the person who created the PKGBUILD (for example by replacing the file on Adobes servers, messing with the connection between your computer and Adobes servers, ...).
But since the URL for the tarball is not for a specific version of the Flash player and adobe has released a new version (188.8.131.526) for Linux, the problem is just that the checksums need to be updated for the new tarball. You can do that by running
updpkgsums in the directory containing the PKGBUILD file. Usually the provider of the tarball would provide checksums himself that you can use to verify
updpkgsums downloaded the correct file or to manually edit them in the PKGBUILD, but Adobe doesn't seem to do that.
FWIW, after running
sha512sums array looks like this on my computer:
If you know a PKGBUILD on the AUR needs to be updated for a new version of the package, make sure to also click "Flag package out-of-date" to notify the maintainer. I've done that for this package already.