What is the ruling on someone who swore by Allah, in a state of anger, not to speak to a person, then spoke to them on the same day, while not remembering the intention of the oath? And is expiation by fasting permissible in this case, given the inability to afford feeding or clothing?

You are required to offer expiation for your oath, as you broke your promise not to speak to the girl. Your anger has no bearing on this, unless it reached the point of losing consciousness, which did not occur. The expiation for an oath is to feed ten needy persons, or clothe them, or free a slave. If you cannot afford this, then you must fast for three days.