Rabbi Yisrael Salanter writes about this story in Or Yisrael, letter #10. He notes the word "chazar - returned", but also that it was said that Rabbi Aqiva hadn't studied Torah until he was 40. In his depiction, Akiva starts studying and feels like he's not getting anywhere. About to give up, others show him the water on the stone, letting him draw the conclusion about carving. Akiva learns that these things take time, and giving up that soon was premature.
באבות דר׳ נתן פ״ו וז״ל: מה היתה תהילתו של ר׳ עקיבא , אמרו בן ארבעים שנה היה ולא שנה כלום. פעם אהת היה עומד על פי הבאר, אמר "מי חקק אבן זו?" אמרו לו "המים שתדיר נופלים עליה בבל יום." אמרו לו "עקיבא, אי אתה קורא אבנים שחקו מים." מיד היה דן ר' עקיבא קו״ח בעצמו, מה רך פסל את הקשה , ד"ת שקשה בברזל, על אחת כמה וכמה שיחקקו את לבי שהוא בשר ודם! מיד חזר ללמוד תורה כו׳ ע״ש.
כי ר׳ עקיבא בתחילת לימודו , באין רושם מוצאת אצלו; אמר נואש לנפשו, והראו לו סתירת הדבר משפיכת המים על האבן ׳ אשר לא יראה מאומה השתנות בהאבן, ועכ׳׳ז המחקרים בשכלם גוזרים, כי נעשה רושם נעלם מהרגש אנושי, היא הסיבה לחקיקת האבן ׳ מריגוי המים ברבות הימים , מחתקבצות הרשימות הבלי נרגשות:
Avos DeRabbi Natan ch. 6: What was R' Aqiva's beginning? They said he was 40 years old and hadn't studied anything. One time he was standing by the mouth of a spring. He said, "Who engraved into this stone?" They told him, "The water that constantly falls on it every day." They said to him, "Aqiva, do you not encounter [other] stones that were engraved by water?" Immediately, R' Aqiva made an a fortiori argument about himself. "If something soft can carve something hard, Torah thoughts which are as 'hard as iron' all the more so it can engrave itself one my heart which of flesh and blood!" Immediately he returned to his Torah study. (Etc... See there.)
Rav Yisrael explains the quote:
When Rabbi Aqiva was starting his learning, there was no impression left on him. He said he felt desperately alone. So they showed him a contradiction to this idea [his frustration and desperation -mb] from the pouring of water upon stonem which does not appear capable of changing the stone. even so those who seek with their minds conclude that it makes an impression that is hidden from human perception. It [that invisible impression] is the reason for the engraving in the stone from all the water in the multitude of water in the accumulation of unfelt impressions.
To Rabbi Yisrael, the story is one of not giving up hope when one's learning just isn't working. We might think nothing is happening, but we are wrong. Every exposure is an impression, and added up over time change is real.
(The advertising industry is built on abusing this same idea. No one thinks someone will choose heart medication or a political candidate because they saw an ad once. But many such impressions, over time, does end up swaying attitudes.)
Notice that R' Yisrael's message requires understanding that R' Aqiva didn't once learn in his youth and pick it up a long time later when he saw the stone. Rather, the stone was a step within his starting to learn. Just not the very first step.