1) THE REASON TO TREAT THE SECOND DAY OF TISHREI LIKE A YOM TOV
QUESTION: The Gemara asks why the Rabanan wanted to establish a day of Yom Tov on the third of Tishrei during the times of the Beis ha'Mikdash, when there already was a Yom Tov on that day, Tzom Gedalyah (which was a Yom Tov during the times of the Beis ha'Mikdash). The Gemara first suggests that perhaps they established another Yom Tov on that day so that the preceding day, the second of Tishrei, would also be treated like a Yom Tov (and be Asur b'Hesped uv'Ta'anis). The Gemara responds that it still would be unnecessary to make a new Yom Tov for that reason, because the second day of Tishrei is already Asur b'Hesped because it is the day which follows Rosh Chodesh.
Why does the Gemara ask that the second day of Tishrei is already Asur b'Hesped because it follows Rosh Chodesh? The Gemara should ask that it is Asur b'Hesped because it itself is a Yom Tov -- it is the second day of Rosh Hashanah (or at least the second day of Rosh Chodesh).
(a) The RITVA answers that the reason why the Gemara says that the second of Tishrei is Asur b'Hesped for it is the day after Rosh Chodesh (and after Rosh Hashanah) is because in Yerushalayim the second of Tishrei was not observed as Rosh Hashanah or as a possible day of Rosh Chodesh. The Gemara wants to teach that even in Yerushalayim itself nothing was gained by establishing a new Yom Tov, because the second of Tishrei was already Asur b'Hesped even in Yerushalayim (where there is only one day of Rosh Hashanah) because it is the day after Rosh Chodesh.
(b) The CHESHEK SHLOMO explains that the Gemara here supports the opinion of RASHI, who writes earlier (18a, see Insights there) that only one day of Rosh Hashanah was observed even outside of Eretz Yisrael during the times that the Sheluchim were sent there. Therefore, the only reason why the second of Tishrei is Asur b'Hesped is because it is the day after Rosh Chodesh and Rosh Hashanah.
2) WHY THE THIRD OF TISHREI NEEDS "CHIZUK"
QUESTION: The Gemara concludes that the Rabanan established a new Yom Tov on the third of Tishrei even though there was already a Yom Tov on that day (Tzom Gedalyah; see previous Insight), in order that the second of Tishrei be treated like a Yom Tov (and be Asur b'Hesped uv'Ta'anis) by virtue of the fact that it precedes a Yom Tov d'Rabanan. The Gemara says that even though the second of Tishrei follows and precedes two other Yamim Tovim, as it is the day after Rosh Chodesh (and Rosh Hashanah) and it is the day before Tzom Gedalyah, it would not have been Asur b'Hesped without a new Yom Tov d'Rabanan because those Yamim Tovim (Rosh Chodesh/Rosh Hashanah and Tzom Gedalyah) are Yamim Tovim mid'Oraisa and mi'Divrei Kabalah which do not need "Chizuk," and thus there is no need to make the day before or after them Asur b'Hesped. Now that the Rabanan established a new Yom Tov d'Rabanan on the third of Tishrei, that Yom Tov d'Rabanan needs Chizuk and thus the second of Tishrei is Asur b'Hesped.
The element of Chizuk in prohibiting eulogies and fasts on the day before the Yom Tov is so that people will not be lenient with the Yom Tov which the Rabanan made. Here, though, the people will not be lenient with the Yom Tov which the Rabanan made on the third of Tishrei, because that day is already a Yom Tov mi'Divrei Kabalah which people are very careful to observe! Why, then, was there a need to give that day extra Chizuk and make the preceding day Asur b'Hesped? (The answer cannot be that it needs Chizuk for when the Beis ha'Mikdash is not standing, at which time Tzom Gedalyah is no longer a Yom Tov but a day of fasting, because the Gemara says that after the Churban all of the Yamim Tovim d'Rabanan mentioned in Megilas Ta'anis are annulled. Moreover, even if the Yamim Tovim in Megilas Ta'anis are not annulled after the Churban, the Gemara should have given this as an answer if it was true.)
(a) The PNEI YEHOSHUA suggests that the Rabanan instituted that Hesped is prohibited on the day before a Yom Tov d'Rabanan even when that Yom Tov was also a Yom Tov d'Oraisa because of "Lo Plug" -- in order not to differentiate between Yamim Tovim d'Rabanan. (This answer needs further clarification, because it implies that the Rabanan made the third of Tishrei a new Yom Tov d'Rabanan simply to make the day before it Asur b'Hesped. If, however, the day itself (the third of Tishrei) does not need Chizuk, then why did they establish on that day a new Yom Tov merely to prohibit the day before?)
(b) The reason the third of Tishrei needs Chizuk is not because of a concern that people will not celebrate that day. Rather, it needs Chizuk so that people will remember the miracle for which the new Yom Tov was made. Even though the observance of the Yom Tov of the third of Tishrei does not need Chizuk because it is a Yom Tov mi'Divrei Kabalah, it nevertheless needs Chizuk so that people will remember the miracle that happened for which the Rabanan enacted a new Yom Tov.