Jesus did indeed baptize, just as the scripture states.
In John 3:22, it is written,
After these things, Jesus came, and his disciples, into the land of Judea, and he stayed with them there, and he baptized.
μετὰ ταῦτα ἦλθεν ὁ Ἰησοῦς καὶ οἱ μαθηταὶ αὐτοῦ εἰς τὴν Ἰουδαίαν γῆν καὶ ἐκεῖ διέτριβεν μετ᾽ αὐτῶν καὶ ἐβάπτιζεν
However, Jesus himself did not baptize. Jesus baptized because his apostles baptized in his name.
In John 4:1-3, it is written,
1 Therefore, when the Lord knew that the Pharisees heard that Yeshu'a makes and baptizes more disciples than Yochanan 2 (although Yeshu'a himself did not baptize, but rather, his disciples), 3 he left Judea and departed again into Galil.
1 ὡς οὖν ἔγνω ὁ Κύριος ὅτι ἤκουσαν οἱ φαρισαῖοι ὅτι Ἰησοῦς πλείονας μαθητὰς ποιεῖ καὶ βαπτίζει ἢ Ἰωάννης 2 καίτοιγε Ἰησοῦς αὐτὸς οὐκ ἐβάπτιζεν ἀλλ᾽ οἱ μαθηταὶ αὐτοῦ 3 ἀφῆκεν τὴν Ἰουδαίαν καὶ ἀπῆλθεν πάλιν εἰς τὴν Γαλιλαίαν
To act or speak "in the name of" another means to effectively be acting or speaking as though that individual himself was acting or speaking.
There’s an example of this in the Book of Esther. In Est. 3:12, the author writes that a group of scribes were instructed to write a letter by Haman. This letter was addressed to the governors and rulers of the provinces. The letter is said to be written “in the name of King Achashverosh” (בְּשֵׁם הַמֶּלֶךְ אֲחַשְׁוֵרֹשׁ). We know for certain that King Achashverosh did not write the letter himself; rather, his scribes did, being instructed by Haman. However, the king’s scribes wrote the letter with the authority of the king himself. Thus, it was as though the king himself had written it.
Thus, when the apostles baptized "in the name of Jesus," the act of baptism is as though Jesus himself performed it. This is important because individuals baptized became slaves of Jesus Christ.
In the Mishneh Torah, Moshe ben Maimon (רמב"ם) describes the practice of immersing (baptizing) converts and slaves (servants).
Just as we circumcise and immerse ("baptize") the converts, likewise we circumcise and immerse the slaves who are acquired from the Gentiles for the sake of servitude… Therefore, his master needs to overcome him in water, until he arises and he is under his servitude.
כְּשֵׁם שֶׁמּוֹלִין וּמַטְבִּילִין אֶת הַגֵּרִים, כָּךְ מוֹלִין וּמַטְבִּילִין אֶת הָעֲבָדִים הַנִּלְקָחִים מִן הַגּוֹיִים לְשֵׁם עַבְדוּת… לְפִיכָּךְ צָרִיךְ רִבּוֹ לְתָקְפוֹ בַּמַּיִם, עַד שֶׁיַּעֲלֶה וְהוּא תַּחַת שִׁעְבּוּדוֹ
For more information on "baptism in the name of Jesus," see my blog.
Moshe ben Maimon, Sefer Kedusha, Hilkhot Issurei Bi'ah, Chapter 13, Halakha 11