Whose account (Konto) is it that the money is coming from when the machine (die Maschine) disburses it to somebody?
What's meant here is not the account or cash cartridge of the machine, but the account of the person using the ATM to withdraw money. That person is refered to by "einem" and "seinem" in the original sentence.
I guess what could be confusing here is mainly the use of "einem", which in this sentence is basically the dative case of the indefinite pronoun "man".
From Wiktionary:
Das Indefinitpronomen man hat [...] keine weiteren [deklinierten] Formen, sodass in anderen Fällen als dem Nominativ Singular auf das Indefinitpronomen ein zurückgegriffen wird, namentlich einem im Dativ und einen im Akkusativ.
Jemandem etwas auszahlen means to disburse something to somebody.
The possessive "seinem" in turn refers back to "einem" - the account belongs to the person who was mentioned earlier.