What does '{I gave one to} [the] both of you' mean?

The correct way to say that you gave one to each person is "I gave one to each of you". Saying "I gave one to both of you" really means that one item is being shared between two people. Contextually though, if you prefer to say "both of you" it is possible to convey how many items were given through the surrounding context.

"The both of you" is a more collective expression, usually used to group two people together as one entity, often in anger. eg "There will be a scolding given to the both of you".

It is actually never correct to say, "the both of you" This is colloquial, from spoken Irish and Scottish and therefore, to be correct, you should say, "the both of yous"! However, it is colloquial and should only be used in very informal speech. "Yous" is (very informal and in speech only) Scottish/Irish plural of, "you".