What do I do with my KeyCard?

You should print the KeyCard and/or save the PDF to an offline storage device such as a USB stick. The KeyCard should be kept in a safe place, such as a bank vault or home safe.

Should I write my wallet passphrase on my KeyCard?

No! BitGo's approach to security depends on your passphrase and KeyCard being separate from each other. If your wallet passphrase is on your KeyCard, then anyone who gains access to your KeyCard will know your wallet.