I'll be talking about digital signatures which is the basis for such things. I assume basic understanding of asymmetric cryptography and hashing.
Basically, you hash the content you want to verify with a secure hashing function and encrypt the value with your private key. You can now append this encrypted value to the content or just release it alongside it.
To now verify this content they can use your public key to decrypt your signature and get the original hash value, and compare it to their own. To get that, they just need to hash the content themselves with the same function.
So by signing their videos with the white house private key and publishing their public key somewhere, you can verify the video's authenticity like that.