The Gemara cites a dispute between Rabbi Yochanan and Reish Lakish regarding the status of the Bechoros that that were born in the Wilderness. Rabbi Yochanan maintains that the Bechoros that were born in the Wilderness were sanctified and Reish Lakish posits that they were not sanctified. The Gemar concludes that both Rabbi Yochanan and Reish Lakish both agree that the Bechoros that were born in the Wilderness were sanctified. However, Rabbi Yochanan maintains that the sanctity of the Bechoros was enduring whereas Reish Lakish posits that the sanctity did not continue.
Further on (5a) the Gemara states that Rabbi Yochanan saw in a dream support for his opinion. It is said (Bamidbar 3:13) לי יהיו, they shall be Mine [- I am HaShem], which is interpreted to mean בהוייתן יהיו, they remain in their sanctified state. Reish Lakish, however, reasons that it is said (Shemos 13:5) והיה כי יביאך, and it shall come to pass when HaShem shall bring you to the land… and then it is said (Ibid verse 12) והעברת, then you shall set apart… and the juxtaposition of these verses teaches us that the Bechoros that were born in the Wilderness did not continue to have sanctity. Rabbi Yochanan, however, explains the juxtaposition of the verses like the academy of Rabbi Yishmael, who taught: perform this mitzvah so that in its merit you will enter into Eretz Yisroel.
What is so unique about the mitzvah of Bechor that observing the mitzvah made the Jewish People worthy of entering Eretz Yisroel? Furthermore, the Sifri (beginning of Ki Savo) states that by observing the mitzvah of Bikkurim the Jewish People would be worthy of entering Eretz Yisroel. The difficulty with this statement is that the mitzvah of Bikkurim did not even apply until they entered the Land (see Panim Yafos and Chasam Sofer there for answers to this question). It would seem from the Sifri and our Gemara that there is a connection between the mitzvah of Bechor and the mitzvah of Bikkurim, as the performance of both mitzvos makes us worthy of entering the Land.
Perhaps the idea is that the essential reason for the mitzvah of Bechor is because HaShem killed all the firstborn Egyptians. The first of something is always deemed to be the strongest, so HaShem was demonstrating to the Egyptians that they could not rely on the strongest amongst them to save them. In a similar vein, when one brings Bikkurim, the first of his fruits, to the Bais HaMikdash, he is declaring that he is totally subjugated to HaShem’s will. Bringing the first of the fruits to the Kohen reflects the idea that he is submitting himself to the leader of the Jewish People and negating his own strength and efforts in cultivating the land.
The meaning of the Sifri regarding Bikkurim and the Gemara regarding Bechor is that in order to enter Eretz Yisroel, one must humble himself first. The Zohar states that for this reason the Land was referred to in Scripture as ארץ כנען, because the word כנען is associated with the word הכנעה, humility. Thus, it is specifically the mitzvos that are connected to the first, Bechor and Bikkurim, that allow us to merit entering Eretz Yisroel, as by submitting our “firsts” to HaShem, He will grant us Eretz Yisroel, of which it is said (Devarim 11:12)ארץ אשר ה’ אלוקיך דורש אותה, תמיד עיני ה’ אלוקיך בה מראשית השנה ועד אחרית שנה, a land that HaShem, your G-d, seeks out; the eyes of HaShem , your G-d, are always upon it, from “the beginning” of the year to the year’s end.