It states (45:27) לָש ֵׂאת יוֹסֵף שָׁלַח ֶׁראֲש הָע ֲגָלוֹת אֶת וַיַּרְא אֲבִיהֶם יַע ֲקֹב ַרוּח וַתְּחִי אֹתוֹ, "He saw the עֲגָלוֹת, wagons, that Yosef had sent to carry him, and the spirit of their father Yaakov was revived." Rashi notes that it should really state הָע ֲגָלוֹת, the wagons that Pharaoh sent, because Pharaoh sent the wagons, and not Yosef (see 45:19).
Rather, the pasuk states יוֹסֵף שָׁלַח ֶׁראֲש הָע ֲגָלוֹת, that Yosef sent the עֲגָלוֹת, which alludes to the ערופה עגלה (the calf that is killed in the field, to atone for a murder that occurred, when the identity of the murderer isn't known). This was the final sugyah that Yosef had studied with his father before they departed. He sent this information with his brothers as a sign that he was indeed Yosef. So, העגלות doesn’t mean wagons, rather the calf, the last subject Yosef and Yaakov studied together.
The Beis Yisrael noted that it states about the ערופה עגלה (Devarim 21:1) נֹפֵל ּ...בַּאֲדָמָה חָלָל יִמָּצֵא כִּי ה ִ כָּהוּ מִי נוֹדַע לֹא בַּשָּׂדֶה, "If a slain person be found in the land which Hashem, your G-d is giving you to possess, lying in the field, the identity of the slayer not being known..." Yosef was hinting to his father that he couldn't know who sold him, and who hurt him. ה ִ כָּהוּ מִי נוֹדַע לֹא, it is not known who harmed him. Therefore, Yaakov shouldn't be angry with any of his children. Although it appears that the brothers sold him, it will never be known, because everything is from Hashem.
It states (45:4) אֲח ִיכֶם יוֹסֵף אֲנִי ...אֶח ָיו אֶל יוֹסֵף וַיֹּאמֶר מ ִצְר ָיְמָה אֹתִי מְכ ַרְת ֶּם ֶׁראֲש, "Yosef said to his brothers, 'Please come closer to me, and they drew closer. And he said, 'I am your brother Yosef, whom you sold into Egypt.'"
Rashi writes, מהול שהוא להם הראה, he showed his brothers that he had a bris milah. The Rishonim ask how a bris milah was proof that he was Yosef, their brother. Some goyim had milah, as we find reference to מהול גבעוני ...מהול ערבי, Arabs or Givonim who have milah.
The Yismach Yisrael ('ב אות) answers that the Midrash (Bereishis Rabba 53:14) states, 'ר אמר עיניהם את מאיר ה"שהקב עד סומים בחזקת הכל בנימין, "Everyone is like the blind, before Hashem enlightens their eyes. As it states (Bereishis 21:19) עֵינֶיהָ אֶת א ֱלֹק ִים וַיִּפְקַח, "Hashem opened her eyes."
The Yismach Yisrael explains that sometimes a person looks for something he lost, and then he finds it right next to him! The same is with every yeshuah that a person needs. He looks all over for the salvation, which might be right before him, but Hashem closed his eyes, and he didn’t see it initially.
(I heard of an older bachur from Bnei Brak who tried very hard to get married. He even once traveled to Montreal to find his shidduch. He ended up becoming engaged to a girl who lived in the same building where his parents lived.)
There is an amount of time set for one to suffer yesurim. Until that time period passes, he can't know about the salvation that is prepared for him. Sometimes, the salvation he needs is right before his eyes. When the term of his suffering ends, Hashem will open his eyes, and he will immediately find the salvation that he seeks.
Yosef said to his brothers, "From the day you sold me, you regretted what you did, and you were prepared to pay all the money in the world to free me and to bring me back to our father. You also made hishtadlus to find me. When you came to Mitzrayim, each of you entered from another gate, because you were hoping to find me. But then you were in Mitzrayim, standing before me. Yet, the idea never crossed your mind that perhaps the person who stands before you is none other than Yosef, your brother! Everyone in Mitzrayim knows that I was once a slave, who became the second to the king, but you didn't suspect that I am Yosef. When the moment came, Hashem opened your eyes, and you knew that I am Yosef, your brother." Yosef told them (45:12) עֵינֵיכֶם וְה ִנֵּה רֹאוֹת, now your eyes see, but before you didn't see because Hashem concealed my identity from you.
This was the reason why when Yosef showed them his bris milah, there was no question anymore. They didn't wonder that perhaps he was an מהול גבעוני ...מהול ערבי. When the time arrived, there were no more doubts. They discovered that what they were looking for was right in front of them!
The Rebbe of Tchechinov zt'l (Divrei Simchah) says a similar vort. When the brothers stood before Yosef, they felt that they were in a very dangerous situation. They were afraid for their lives, for their future. But only three words were between them and their salvation. Those three words were (45:4) אֲנִי אֲח ִיכֶם יוֹסֵף, "I am your brother, Yosef."
This is a lesson we can take when experiencing tzaros. We think that there is no solution, or that salvation is far away, but sometimes it is right there. With just three words, or fewer, everything can turn around.
Sifsei Gavriel (from Reb Gavriel Gestetner zt'l, rav of Sambathali, p.180) explains the words we say in the tefillah ישועה מצמיח, that Hashem sprouts forth salvation. When one plants a seed, it decomposes, and then it begins to grow. The same is with a yeshuah. When it seems there is no more hope, that is when Hakadosh Baruch Hu brings forth the salvation.
Yosef told his brothers (45:7) א ֱלֹק ִים וַיִּשְׁלָח ֵנִי גְּדֹלָה לִפְל ֵיטָה לָכֶם וּלְהַח ֲיוֹת בָּאָרֶץ שְׁא ֵרִית לָכֶם לָשׂוּם לִפ ְנֵיכֶם, "Hashem sent me before you to make for you a remnant in the land, and to preserve it for you for a great salvation."
The Maor v'Shamesh asks why the words בָּאָרֶץ שְׁא ֵרִית לָכֶם לָשׂוּם, "To make you a remnant in the land," are necessary?
It is because it states (Michah 7:18) כָּמוֹךָ ל"אֵ מִי נַחֲל ָתוֹ לִשְׁא ֵרִית ַׁעפ ֶּש עַל וְעֹבֵר ע ָוֹן נֹשֵׂא, "Who is like Hashem, Who forgives iniquity and passes over the transgression of the remnant of His heritage?"
Chazal (Rosh Hashanah 17b) say that the pasuk is saying that Hashem forgives aveiros of those who are נַחֲל ָתוֹ ִשְׁא ֵרִית, who feel like remnants, which means they are humble.
Yosef explained to his brothers וַיִּשְׁלָחֵנִי לִפ ְנֵיכֶם א ֱלֹק ִים, Hashem wanted me to be king, and that you should humble yourselves before me, בָּאָרֶץ שְׁא ֵרִית לָכֶם לָשׂוּם, so you will attain the trait of שְׁא ֵרִית, humility to atone for your aveiros. And then in the next world and in this world, גְּדֹלָה לִפְל ֵיטָה לָכֶם וּלְהַח ֲיוֹת, it will be good for you, for a great salvation.