The Talmud: One who wishes to be a Chossid (“one who goes beyond the letter of the law, and who strives to conduct himself in the best possible way out of love for Hashem”), should fulfill the words of [Pirkei] Avos! Commentaries explain (and the Rebbe constantly emphasizes) that the entire collection of teachings found in Pirkei Avos represent “milsah d’chassidusah” – matters of pious conduct. The Alter Rebbe writes in his Siddur, “It is customary to say Pirkei Avos, one chapter on each Shabbos between Pesach and Shavuos, at Mincha ... And some have the custom to do so through all Shabbosim of the summer”.
Midrash Shmuel: One of the reasons for this practice is because the Torah is given on Shavuos, and one who studies Torah must first know the approach to its study and how he should conduct himself subsequent to its study. It was therefore established to study Pirkei Avos in order that the heart of every Jew will be prepared to accept the Torah wholeheartedly, and to conduct himself with the Torah in a lawful and proper manner.
Additionally, since nature gains strength so-to-speak during the summer season, all people engage in matters that concern bodily health and fitness in the summertime. Now, that may be appropriate for those who conduct themselves according to nature alone; the Jewish nation however, conducts itself according to the Torah. We therefore spend this time involved in the health and fitness of the soul (via studying Pirkei Avos and implementing its directives). This has the natural effect of ensuring a healthy body too (in the manner of preventative medicine).
The Rebbe adds: Since we are concerned with bodily health during the summer, we must make certain that it does not lead to “Yishurun grew fat and kicked [i.e., the Jewish people grew prosperous and rebelled]”. We therefore study Pirkei Avos, which deals with character refinement and arouses the Jewish people to pursue goodly qualities and to step beyond the call of duty. Moreover, the effect of saying Pirkei Avos throughout the summer months is also continued into all the Shabbosim (and thereby into all of the weekdays) of the winter season too!