The Rav asks: why did the Torah mix with the joy of Pesach the sadness of the Sefirah period? Even if one would say that this mourning is on a rabbinic level, obviously our rabbis were connecting this mourning to the Omer since the Omer lends itself to this.
Rabbi Soloveitchik answers this through an analysis of the pesukim that describe the connection between Pesach and the Omer in Sefer Vayikra. It says:
ד אֵלֶּה מוֹעֲדֵי יְקֹוָק מִקְרָאֵי קֹדֶשׁ אֲשֶׁר תִּקְרְאוּ אֹתָם בְּמוֹעֲדָם:
ה בַּחֹדֶשׁ הָרִאשׁוֹן בְּאַרְבָּעָה עָשָׂר לַחֹדֶשׁ בֵּין הָעַרְבָּיִם פֶּסַח לַיקֹוָק:
ו וּבַחֲמִשָּׁה עָשָׂר יוֹם לַחֹדֶשׁ הַזֶּה חַג הַמַּצּוֹת לַיקֹוָק שִׁבְעַת יָמִים מַצּוֹת תֹּאכֵלוּ:
ז בַּיּוֹם הָרִאשׁוֹן מִקְרָא קֹדֶשׁ יִהְיֶה לָכֶם כָּל מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ:
ח וְהִקְרַבְתֶּם אִשֶּׁה לַיקֹוָק שִׁבְעַת יָמִים בַּיּוֹם הַשְּׁבִיעִי מִקְרָא קֹדֶשׁ כָּל מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ: פ
ט וַיְדַבֵּר יְקֹוָק אֶל מֹשֶׁה לֵּאמֹר:
י דַּבֵּר אֶל בְּנֵי יִשְׂרָאֵל וְאָמַרְתָּ אֲלֵהֶם כִּי תָבֹאוּ אֶל הָאָרֶץ אֲשֶׁר אֲנִי נֹתֵן לָכֶם וּקְצַרְתֶּם אֶת קְצִירָהּ וַהֲבֵאתֶם אֶת עֹמֶר רֵאשִׁית קְצִירְכֶם אֶל הַכֹּהֵן:
יא וְהֵנִיף אֶת הָעֹמֶר לִפְנֵי יְקֹוָק לִרְצֹנְכֶם מִמָּחֳרַת הַשַּׁבָּת יְנִיפֶנּוּ הַכֹּהֵן:
טו וּסְפַרְתֶּם לָכֶם מִמָּחֳרַת הַשַּׁבָּת מִיּוֹם הֲבִיאֲכֶם אֶת עֹמֶר הַתְּנוּפָה שֶׁבַע שַׁבָּתוֹת תְּמִימֹת תִּהְיֶינָה:
טז עַד מִמָּחֳרַת הַשַּׁבָּת הַשְּׁבִיעִת תִּסְפְּרוּ חֲמִשִּׁים יוֹם וְהִקְרַבְתֶּם מִנְחָה חֲדָשָׁה לַיקֹוָק:
The Rav notes that despite the fact that the Omer is brought on the second day of Pesach, Pesach and the Omer are presenting separately in the verses. Not only that but the word used by the Karban Omer and the counting of the Omer is מִמָּחֳרַת הַשַּׁבָּת it should begin on the day after Shabbos. Some of you might know that this was the source for the famous argument between the Perushim and Boethusians. The Boethusians read מִמָּחֳרַת הַשַּׁבָּת literally as the day after Shabbos and always started the Sefiras HaOmer on the day after Shabbos following Pesach so that Shavuos was always a Sunday. The Perushim who represent rabbinic Judaism hold that here Shabbos means a day we do not work, namely the first day of the Yom Tov of Pesach and not the day of Shabbos so Shavuos can fall out on any day of the week. This was such a strong argument that on the day that the Perushim finally defeated the Boethusians they made a holiday. Why is it that the Torah chose to call the start of the Omer as מָּחֳרַת הַשַּׁבָּת instead of directly connecting it to Pesach?
The Rav answers that this is because the Pesach and the Omer represent two opposite extremes. Pesach represents the supernatural, Hashem's open revelation and intercession into Jewish history with the Yetzias Mitzrayim. This is what we celebrate on the seder night. But as we know, Hashem cannot always be so openly involved in history. In order for humanity to be endowed with free will and the ability to make independent choices, Hashem must operate behind the veil of a natural world. This is represented by the Omer. When Benay Yisrael arrive in Israel, they will plow and plant the soil and then grow their produce through natural means. After harvesting their first crops, they are commanded to bring the Karban Omer the barley from this natural harvest. This is the reason why the Pesach and Omer are presented separately despite the fact that they coincide on the calendar. The Pesach celebrates the God of revelation while the Omer commemorates the God of natural events. It is also for this reason that the Omer is brought מִמָּחֳרַת הַשַּׁבָּת. Shabbos represents the natural order of creation. In six days, Hashem created the world and on the seventh he rested. It is after Shabbos that we bring the Omer.
The Omer then has a certain sadness to it. After experiencing the great joy and revelation of the Pesach holiday. We have to go back to a normal existence. It is back to work and back to school.
The challenge of the beginning of the Omer is to bring the Karban Omer, to recognize that Hashem is just as involved in the natural as he is in the supernatural, only he is hidden from view. Hashem is always there. Only he hides behind the veil of nature and history. For brief fleeting moments he shows himself in all of his glory like at the splitting of the Yam Suf. We remember this and use these moments of inspiration to see Hashem throughout the rest of the year in natural events. It is this that we commemorate through the counting of the Omer.