How can one atone for breaking the fast in Ramadan due to thirst and hunger without the family's knowledge during puberty, given that the number of days is not remembered?

You must make up the days you missed in Ramadan. If you delay making them up without a valid excuse until the next Ramadan arrives, then in addition to making them up, you owe an expiation for the delay, which is feeding one poor person for each day. The expiation for delay does not multiply with the number of Ramadans delayed. If you do not know the exact number of days you missed, then make up what you believe, most likely, will absolve you of your obligation.