Almost all the yeshiva high schools strongly push all the graduates to spend their year after high school in one of these programs. I am curious as to why there is this emphasis? What do rashei yeshiva, parents and graduates hope to gain from the Israel experience that might not otherwise be accomplished similarly or better by attending a yeshiva program in their own city or an out-of-town yeshiva in the U.S.?
(Disclaimer: I do not work for the Yeshiva high schools, nor for the Yeshivot in Israel, so I can't say that this is their reason. All I can say is that this could be a reason.)
The simple explanation is that Israel is better suited for Torah study.
כי סליק רבי זירא קם בשיטתיה דרבי אילא קם רבה בשיטתיה דרבי זירא אמר רבי זירא שמע מינה אוירא דארץ ישראל מחכים
When R. Zera went up [to Palestine] he adopted the principle of R. Elai. R. Zera said: From this one may deduce that the climate of the land of Israel makes one wise. (Soncino translation)
(There are various other Talmudic statements about the superiority of Israeli Torah study over Babylonian Torah study, but most of those would probably have to be interpreted within the specific context of the Talmudic Era.)
Additionally, Israel has various benefits relating to mitzvos other than Torah study. In fact, R. Hayim David Halevi has an interesting responsum. A student wanted to study in Israel and his parents wanted him to remain in the Diaspora. The rabbi responded at length that the student is not obligated to listen to his parents in this matter, and he spent most of the responsum explaining that the reason is that there is a mitzvah to live in Israel.
Shu"t Aseh Lecha Rav 1:17
כל ההקדמה הארוכה הזאת באה כדי להכין אותך לקבל ההלכה הפשוטה והברורה שאין עליה חולק בלב שקט כי כך היא הלכה שאין אתה רשאי לשמוע לקול הוריך כאשר רצונם מתנגש עם צו התורה והלכה פסוקה היא מי שאמר לו אביו לעבור על דברי תורה בין שאמר לו לעבור על מצות עשה או לבטל מצות עשה אפילו של דבריהם הרי זה לא ישמע לו שנאמר איש אמו ואביו תיראו ואת שבתותי תשמורו כולכם חייבים בכבודי (הרמב"ם הלכות ממרים פרק ו' הי"ב ומקורה ידוע ביבמות ו' ובבא מציעא ל"ב) ועתה התבונן אם בכל מצוות התורה אין הבן רשאי לשמוע לקול אביו כשמצוהו לעבור על אחת מהן כל שכן וקל וחומר כאשר מצוהו לעבור על מצות ישיבת א"י שהיא שקולה ככל המצוות שבתורה שבודאי שאין הוא רשאי לשמוע לאביו ומבלי להאריך בציטוטין וברשימות של פוסקים אציין לך בקצרה שזו דעת כל גדולי הפוסקים
Our texts are full of praise for the Land of Israel and how Torah study there is of a different quality than in the Diaspora. Maybe the rashei yeshiva and parents who send kids to Israel want them to experience this for a year as they transition into adulthood, as a way to further elevate them and give them an even stronger Torah platform to stand on.
R Dov Lipman (an American rabbi who made alyah in 2004 and served as a Member of Knesset from 2013-2015) brings many of these sources in Coming Home: Living in the Land of Israel in Jewish Tradition and Thought and I selected a few relevant ones to this question in particular
There is no Torah like the Torah of the Land of Israel and there is no wisdom like the wisdom of the Land of Israel. (Bereshit Rabbah 6)
Rabbi Yossi bar Chalafta told his son, Rabbi Yishmael, ‘Do you want to see the Divine Presence in this world? Study Torah in the Land of Israel.’ (Midrash Tehilim 105)
When I come to the Holy Land, I feel clearly that here my heart is more easily aroused to the study of Mussar. In addition, one feels the heavenly assistance in Torah study here.... In our generation, we see with our own eyes how youngsters who study Torah flourish in the Land of Israel. They advance in their studies, they succeed in acquiring knowledge in all of the Talmud, and they achieve much more than others achieve in their Torah education in other lands. (Michtav m'Eliyahu 3:194)
The Kli Yakar explains that when the Torah calls Israel a “good land,” (Bamidbar 13:2) it is referring to spiritual goodness, since "the air of the land itself helps a person attain true good."
I have now seen an interesting answer from a very different angle. R Aharon Leib Steinmann (in his artscroll biography, p. 394) was asked whether it would be a good idea to establish a state-of-the-art yeshiva in the United States and bring in some high-caliber Roshei Yeshivah from Israel to give shiurim, thereby obviating the need for the boys to travel abroad.
He answered he did not like the idea. He felt it was beneficial for boys from the United States and other countries to see firsthand that although the standard of living in Israel may be several notches below the norm abroad, avreichim there are happily learning Torah with great hasmadah and without the luxuries and large houses that are so common in other locales. This, he felt, is a whorthwhile lesson to absorb.
(told by R Chizkiahu Mishkovsky who was present)