The Rashban in one of his responsa (136), writes that one who partakes of the festive meal celebrating a Pidyon Haben is considered to have observed 84 fasts. This concept appears in several other sources, as well, including the Shem Mi’Shimon Pollack (Y.D. 29), the Ta’ameh Ha’minhagim, and the Sedeh Hemed (Rav Haim Hizkiya Medini, 1833-1905). Interestingly, the Sedeh Hemed writes that he was once invited to attend a Pidyon Haben in Jerusalem, and he declined. He was then told that participating in a Pidyon Haben celebration is akin to observing 84 fasts, a notion that he had not known before. The Sedeh Hemed adds that he later found written in books that one who partakes of a Pidyon Haben meal is considered to have observed the 84 fasts of Shobabim – the six-week period (from Parashat Shemot through Parashat Mishpatim) when it is customary to fast and repent for certain sins involving the Berit. (The Kaf Ha’haim Sofer, in Orah Haim 240:6, explains the precise procedure for fasting during the weeks of Shobabim whereby one is considered to have observed 84 fasts.)
The concept of the 84 fasts of Shobabim is alluded to in Yaakob’s blessing to Reuben (Bereshit 49:3), in which he says, "Reuben, you are my firstborn; my strength, and the first of my issue." This means that Reuben was conceived from the first seed that left Yaakob’s body; he did not experience an emission until then. This occurred when Yaakob was 84 years old, and thus the 84 fasts of Shobabim correspond to the remarkable self-control that Yaakob exercised for 84 years. Likewise, the Pasuk in Tehillim (111:9) states, "Pedut Shalah Le’amo Siva Le’olam Berito" ("He sent redemption to His nation; He has established His covenant for eternity"). The word "Pedut" (which is spelled "Peh," "Dalet," "Tav") represents the words "Peh-Dalet Ta’aniyot" – 84 fasts. This alludes to the 84 fasts that God "sent us" so we can earn atonement for sins relating to the Berit ("Siva Le’olam Berito").
The work Osar Pidyon Haben records an incident of a Rabbi who observed the fasts of Shobabim, and his students asked him why he went through the trouble of fasting if he could achieve the same result simply by participating in a Pidyon Haben celebration. He answered that the sources say regarding somebody who partakes of a Pidyon Haben meal that through this Misva he is considered "Ke’ilu" ("as if") he had observed 84 fasts. This refers to the statements in the Talmud where certain improper activities are equated with grave transgressions with the phrase "Ke’ilu." For example, the Talmud teaches that one who grows angry is deemed "Ke’ilu" – as though – he worshipped idols. This formulation appears 84 times in the Talmud, and thus partaking of a Pidyon Haben meal atones for those 84 sins which the Talmud speaks of with the term "Ke’ilu."
It should be noted that Hacham Ovadia Yosef ruled that it suffices to eat cake and the like to fulfill this Misva; it is not necessary to eat an actual meal to reap the enormous benefits of participating in a Pidyon Haben celebration.
Summary: One who takes part in a Pidyon Haben meal is considered to have observed the 84 fasts of the Shobabim period. It suffices to eat cake and similar foods; one does not have to actually eat a full meal at the Pidyon Haben celebration to earn this great reward.