(15) And from the day on which you bring the sheaf of elevation offering—the day after the sabbath—you shall count off seven weeks. They must be complete: (16) you must count until the day after the seventh week—fifty days; then you shall bring an offering of new grain to יהוה.
you shall count fifty days In Biblical Hebrew, the number given always includes both the first and the last days of the count. Similarly, Scripture always means one week when it uses the phrase “on the eighth day” [e.g., 12:3].
The omer is an Israelite unit of dry measurement used during the time of the Temple in Jerusalem, used for dry volume of grains and other goods. The Jewish Study Bible suggests that an omer measures about 2.3 liters. Apparently it is also equivalent to 43.2 eggs or about 3.5-4 lbs (how much dough you need to take out an offering to make it officially challah).
https://en.wikipedia.org/wiki/Omer_(unit)
Sefer HaChinuch: Composed in Middle-Age Spain (c.1255 - c.1285 CE). The Sefer ha-Chinuch (Book of Education), is a work which systematically discusses the 613 commandments of the Torah. It was published anonymously in 13th century Spain.
ומפני כן, כי היא כל עקרן של ישראל ובעבורה נגאלו ועלו לכל הגדלה שעלו אליה, נצטוינו למנות ממחרת יום טוב של פסח עד יום נתינת התורה להראות בנפשנו החפץ הגדול אל היום הנכבד הנכסף ללבנו כעבד ישאף צל, וימנה תמיד מתי יבוא העת הנכסף אליו שיצא לחרות, כי המנין מראה לאדם כי כל ישעו וכל חפצו להגיע אל הזמן ההוא. וזהו שאנו מונין לעמר, כלומר, כך וכך ימים עברו מן המנין ואין אנו מונין כך וכך ימים יש לנו לזמן, כי כל זה מראה לנו הרצון החזק להגיע אל הזמן, ועל כן לא נרצה להזכיר בתחילת חשבוננו רבוי הימים שיש לנו להגיע לקרבן שתי הלחם של עצרת...למה לא נזכיר מעוט הימים הנשארים? לפי שאין לשנות מטבע החשבון באמצעו. ואם תשאל אם כן למה אנו מתחילין אותו ממחרת השבת, ולא מיום ראשון? התשובה כי היום הראשון נתיחד כלו להזכרת הנס הגדול, והוא יציאת מצרים, שהוא אות ומופת בחדוש העולם ובהשגחת השם יתברך על בני אדם, ואין לנו לערב בשמחתו ולהזכיר עמו שום דבר אחר.
Now, for this reason, because it is the main core of Israel, and because of it they were redeemed and rose to distinction - we were commanded to count the days from the morrow after the festive day of Pesach until the day of the giving of Torah; to show with our whole selves our yearning for that sacred day, which our hearts yearn [for] like 'a slave seeks shade' and always tallies when will come the yearned time that he goes out to freedom. For counting shows a persons hopes of deliverance and desire to reach that time. That is why we count the omer, saying, "so many days have passed from the count," and we do not count "so many days remain for us to count," because this shows our great desire to reach Shavuot. Therefore, we don't want to mention at the beginning of our counting the significant number of days remaining to reach the offering of two breads of [Shavuot]. ...And if you shall ask, "If so, why do we begin counting from the day after [Pesach] and not from the first day?" The answer [is that] it is because the first day is entirely dedicated to remembering the great miracle, which is the exodus from Egypt, that is a sign and a proof of the world having been created and of God's - may God be blessed - providence over people. And we may not mix [something else into] its joy and mention anything else with it.
Atara Lindenbaum, Yeshivat Maharat, May 8, 2020
The method of counting backwards, though, helps us re-enact the way the Jewish people felt when they left Egypt. Bnei Yisrael did not know when they would receive the Torah, they did not know how their journey in the desert would continue, and therefore could not possibly count towards anything, since that day was unknown and perhaps unimaginable. As we count only the days that have passed since Pesach, we can imagine how our ancestors felt in the desert as each day passed, as they had no idea what tomorrow would bring. Perhaps they also kept a count of how many days they managed to keep going in the desert, not knowing exactly from where their sources of food, water, and spirituality would come.
The Observant Life: The Wisdom of Conservative Judaism for Contemporary Jews, Martin S. Cohen, Senior Editor
Since the destruction of the Temple, the custom of counting the omer mostly serves to draw a direct line between Passover and Shavuot, a line that can be taken to connect the concept of growth from mere freedom from slavery to the point at which one understands that ultimate freedom comes to human beings through obedience to divine law.
וְזֶהוּ "וּסְפַרְתֶּם לָכֶם"-'לָכֶם' לְעַצְמֵיכֶם. 'לָכֶם'-לְעַצְמֵיכֶם דַּיְקָא, שֶׁכָּל אֶחָד צָרִיךְ לִסְפֹּר סְפִירַת הָעֹמֶר, שֶׁהוּא בְּחִינַת הִתְחַזְּקוּת הַנַּ"ל לְעַצְמוֹ כְּפִי מַה שֶּׁהוּא.
“And you shall count for yourselves” (Vayikra, 23:15)
“For yourselves” actually means - for you. Counting the Omer is a process of strengthening yourself, and each individual must do that for themselves, based on who they are.
Rambam, Moreh Nevuchim (Guide for the Perplexed) 3:43
Shavuot is the time of the Giving of the Torah. In order to honor and elevate this day we count the days from the previous festival until it [arrives], like someone who is waiting for a loved one to arrive, who counts the days by the hours.
(Another perspective...)
Rabbi Aryeh Carmel, Masterplan, Feldheim - p. 205
The Omer on Pesach was from the barley harvest. The offering on Shavuot was of wheat. Barley is mainly food for animals. Wheat is food for human beings. The Torah hints to us that physical independence by itself still leaves man – from the Torah perspective – on the animal level. The counting of the forty-nine days signifies a sevenfold refining process and marks our progress to full human status with our acceptance of the Torah at Sinai, seven weeks after the Exodus.
How to Count
בָּרוּךְ אַתָּה יְיָ אֱלֹהֵֽינוּ מֶֽלֶךְ הָעוֹלָם, אֲשֶׁר קִדְּשָֽׁנוּ בְּמִצְוֹתָיו, וְצִוָּֽנוּ עַל סְפִירַת הָעֹֽמֶר.
Baruch atah Adonai Eloheinu melech ha’olam asher kid’shanu b’mitzvotav vitzivanu al sefirat ha’omer.
Blessed are You, Adonai our God, Ruler of the universe, who sanctifies us with mitzvot, and commands us concerning the counting of the Omer.
After the blessing one recites the appropriate number of the count. After the first six days, one also includes the number of weeks that one has counted. For example:
Hayom sh’losha asar yom, she’hem shavuah echad v’shisha yamim la’omer
Today is 13 days, which is one week and six days of the omer
From Lurianic Kabbalah in the 17th and 18th centuries to attach a mystical meaning to a regular practice of counting.