You should join his class.

I took some of a colleague’s classes while she was overseas. A student from that class is now coming to mine. She said she likes my teaching. She said she told her brother about me, and:

[1] He said you should join his class.

I was confused. Where is his class and why should I join it? I asked her something along those lines, and she said either:

[2] He said, “You should join his class”.

or

[3] He said I should join your class.

One of the rules of changing direct quotations into indirect ones is pronoun changes, especially I and you. Another is that direct quotations are usually indicated in speech by a slight pause before the quoted words. She hadn’t paused, or hadn’t paused long enough. 

Interpreting [1] as an indirect quotation, as I did, gives:

[1’] “He [brother] said you [teacher] should join his [brother’s] class”. 

This is the equivalent of:

[4] He said, “He should join my class”.

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“And then he kissed me”

The textbook illustrates indirect quotation with the song “And then he kissed me”, written by Phil Spector, Ellie Greenwich and Jeff Barry and originally and most famously sung by The Crystals. Direct and indirect quotation can be seen in the pair of lines:

So I whispered “I love you”
And he said that he loved me too

“I love you” are her actual words and are indicated by quotation marks. (Or they should be – they are in the textbook, but not on the website I copied the lyrics from rather than typing them from scratch.) Indirect quotations are typically introduced by the subordinator that, and (hardest for ESL learners) changes of person (usually pronouns) and verb tense. He actually said “I love you, too”. Because she is reporting his words, his I becomes her he and his you becomes her me. The verb tense typically moves back one “time” (sometimes referred to as backshift), in this case from present simple love to past simple loved. But this optional. If she is reporting his words soon after (for example to a friend the next day), she might keep present simple and say “And he said that he loves me too”. If she is reporting his words a long time later (for example to their grandchildren after his death), she would certainly backshift and say loved.