Your parsing is not quite right. Literally, 僕は迷い込んだ夢の中 means "I am in a dream I strayed into." Note that it's basically a full, regular sentence except that the last verb だ is omitted. I can say this because, as a general rule, は is a "topic" marker and thus rarely appears in a relative clause. This is a basic rule of relative clauses:
But before we can relativize them, let's replace the topic particle は with the case particle が. Why? Because relative clauses in Japanese don't contain topics, so you need to use the version with が to turn them into relative clauses.
In other words, this sentence is 僕は夢の中だ ("I am in a dream") with a short relative clause 迷い込んだ.
The subject of 迷い込んだ is 僕, which is the same as the main verb, so it's naturally omitted in this case. You can explicitly specify the subject of the relative clause with が like: 僕は僕が迷い込んだ夢の中だ or 僕は彼女が迷い込んだ夢の中だ ("I am in a dream she strayed into.")
By contrast, 僕が迷い込んだ夢の中 means "middle of the dream I strayed into" (a noun phrase without a main predicate; sentence fragment).