St. Nicodemos has this to say on the topic:
We must keep the fast of the Holy Apostles "not on account of the Holy Apostles, as some say, not on account of the descent of the Holy Spirit, but pre-eminently and principally on account of the preceding days' rest...consequently and according to the concomitant reason, because the divine Apostles were wont to fast and were thus sent out to preach; for it was then (say the Acts, in ch. 13 v. 3) "when they had fasted and prayed, and had laid their hands on them, they sent them away"...
So, we fast in order to make up for the laxity of the fast-free week after Pentecost (although St. Nicodemos seems to kind of contradict himself a bit with his appeal to analogical practice). I should note that St. Nicodemos, in his initial general commentary on the canon in question, emphasizes that we do not fast on account of the Feast (which is joyous), but on account of our sins. Anyway, he says a little ways down (speaking of the Advent fast, the Apostle's fast and the Dormition fast):
But we ought to observe these particular fasts, not with xerophagy*, as in the case of Great Lent, but with wine and olive oil and the eating of fish (i.e., ichthyophagy), except on Wednesdays and Fridays that fall within these fasting periods, and except during the fast of August, on the occasion of which we partake of fish only once, on the feast of the Transfiguration. For notwithstanding the fact that these particular fasts are not ordained by the Apostles**, we are nevertheless in duty bound to observe also the traditions handed down by the Fathers on account of long consuetude, which has the force of a law, according to the sacred and civil laws. And because, according to St. Basil the Great (see his sermon on morals LXX), even in those matters wherein nothing is particularly stated in the Bible, we ought to exhort everyone towards what is best and of the greatest benefit of the soul.
, lit. "dry eating," means (at least for lay people) keeping the full, normal vegan fast, while ichthyophagy
, lit. "fish eating," usually means being able to eat fish, wine and oil.
** St. Nicodemos mentions "the Apostles" not because of the "Apostle's Fast," but because of the canon on which he is commenting, namely, Canon LXIX of the Holy Apostles. These canons were attributed to the Apostles, but were actually written some time around A.D. 400 (it was a normal literary practice in Late Antiquity to ascribe one's writing to prominent personages). At any rate, such does not denigrate these canons' authority in the Orthodox Church, since they were officially adopted into the canonical corpus at the Quinisext Ecumenical Council in 692. Quinisext adopted 85 Apostolic Canons (the full number contained in Eastern manuscripts of the Apostolic Constitutions
,the work in which the canons first appear), but the West adopted only 50 (b/c of Dionysius Exiguus, of course, whose Latin editions account for most of the early differences between Eastern and Western canon law in Late Antiquity and the Medieval period...until Gratian at any rate).