One explanation is that it was a response to Islamic historical accounts of transmission of tradition. As Alexander Guttman writes in "Tractate Abot: Its Place in Rabbinic Literature" (p. 190-191) with my emphasis:
For the Jew living in the Islamic environment, there was no better way
of defending the Torah and its traditional interpretation than by
presenting a Hadith of their own. Of course, it had to be given a
prominent place in Jewish literature, and no more suitable location
could be desired than in the Mishna. The non-legal character of Abot,
as well as its suitability for widespread study and reading by laymen
(as well as scholars), might have been the reason for selecting this
tractate.
To make the defense more effective, the tractate was included in the
liturgy. This took place in geonic times. L. Zunz reports on good
authority that the reading of Abot was customary in Arabic countries
at least as early as the 9th Century and became an established
practice in the 11th Century. Sar Shalom relates that the reading of
Abot on Sabbath after Minha was a customary practice in Baghdad or
Sura. Both Amram and Saadia include in their respective Siddurim the
practice of reading Abot after Minha-service on Sabbath. Thus we see
that the reading of Abot originated in the geonic period in an Islamic
environment. Since, as we have shown, the Hadiths i.e., the
historical stratum were most probably added to Abot in geonic times,
there can be little doubt as to the true reason for its inclusion as a
part of the regular service. Other explanations given in various
literatures are conjectures or aggadot, i.e., weak attempts to explain
an extraordinary addition to the service.
A somewhat similar explanation is that reading Pirkei Avos on Shabbos afternoon is a statement of anti-Karaism. In "The Development of a Waiting Period
Between Meat and Dairy:
9th – 14th Centuries" (p. 4) Steven H. Adams writes as follows, with my emphasis:
Even many minhagim extant today were arguably initiated as a response
to the Karaite movement. For example, many historians agree that the
recital of the 3rd chapter from Mishnat Shabbat, “Bamme Madlikin,” on
Friday evenings following the prayer service was introduced during the
time of the geonim with the intent of reinforcing the rabbinic stance
on having fire prepared before Shabbat, in opposition to the Karaite
view that no fire may be present in one’s home on Shabbat. Similar
arguments have been made for the origins of the custom of reading
Pirkei Avot, the introduction of which traces rabbinic teachings to
Sinai, on Shabbat afternoons.
The footnote to the last sentence states:
Wieder, Jewish Liturgy, p. 350; compare Alexander Guttman, “Tractate
Abot: Its Place in Rabbinic Literature,” JQR 41 (1950), pp. 190-193,
who argues that the rabbinic chain of tradition in Avot was a late
stratum added under the influence of hadıth scholarship. His arguments
do not preclude the existence of anti-Karaite intentions behind Amram
b. Sheshna and Saadya’s inclusion of this portion of
Avot in their liturgies.
As for why, according to these reasons, the custom would be specifically on Shabbos afternoon and during a specific time of year, I would guess that those details are incidental to the actual reason. That is to say that it is probably simply something like Shabbos afternoons in the summer months are a particularly convenient time to read Pirkei Avos. As Rambam wrote in Guide for the Perplexed 3:26 about details of mitzvos:
Those who trouble themselves to find a cause for any of these detailed
rules, are in my eyes void of sense: they do not remove any
difficulties, but rather increase them. Those who believe that these
detailed rules originate in a certain cause, are as far from the truth
as those who assume that the whole law is useless. You must know that
Divine Wisdom demanded it--or, if you prefer, say that circumstances
made it necessary--that there should be parts [of His work] which have
no certain object: and as regards the Law, it appears to be impossible
that it should not include some matter of this kind. (Friedlander translation)