How I can add a "last modified date" to the front of Trello cards?

Someone might have a better idea, but you could create a custom date field called "last modified". Then, use Butler to create rules to say "when this happens, set the custom field date to now".

Not sure if you could create a general trigger to say "when any change happens", but you could create multiple rules for when someone adds a comment, attaches a file, changes the due date etc. it would look like this:

If the purpose of having this field show on the front of the card is to identify cards that haven't been updated in a long time, there's also the "card aging" power up.