Main problem is that your website must use https (and https only) in order to enable this feature. Download files must be served via https too.
We have added explicit support for https-enabled website recently so this might be an option. However, we haven't really looked at the specs recently if anything has changed. As far as I remember, your site needs to be whitelisted by twitter and you might have to go through a registration process which might or might not be open for everybody.
If you can do some research here and tell us we might consider adding twitter player card support