First, download the checksums.txt file and the checksums.txt.sigstore.json file files, for example, with wget:

wget 'https://github.com/charmbracelet/crush/releases/download/v0.38.1/checksums.txt'
wget 'https://github.com/charmbracelet/crush/releases/download/v0.38.1/checksums.txt.sigstore.json'

Then, verify it using cosign:

cosign verify-blob \
  --certificate-identity 'https://github.com/charmbracelet/meta/.github/workflows/goreleaser.yml@refs/heads/main' \
  --certificate-oidc-issuer 'https://token.actions.githubusercontent.com' \
  --bundle 'checksums.txt.sigstore.json' \
  ./checksums.txt

If the output is Verified OK, you can safely use it to verify the checksums of other artifacts you downloaded from the release using sha256sum:

sha256sum --ignore-missing -c checksums.txt

Done! You artifacts are now verified!