In the libretto of the Magic Flute, the phrase "Tempel ein" is often used. E.g. Sarastro says "Führt diese beiden Fremdlinge in unsern Prüfungstempel ein" and Monostatos says "...bald dringen wir in Tempel ein". What does the addition of 'ein' do, other than to make it rhyme with the next line, and how does this work?
Welcome to the wonderful world of german separable verb prefixes :-) "Tempel ein" isn't a phrase but the result of separating the verb's prefix from the main part of the verb.
In your examples, the complete verb is "einführen" resp. "eindringen". In both cases, you can separate the prefix "ein-" from the rest of the verb; and the difficult thing is that the prefix may be found at the very end of the sentence, so you even know the complete verb only after reading the complete sentence. I've put the verb in your examples in bold letters:
"Führt diese beiden Fremdlinge in unsern Prüfungstempel ein" - "Take these two strangers into our temple of trial"
...bald dringen wir im Tempel ein". "...soon we will be within the Temple!"
In German, there are also inseparable verbs, and verbs that can exist as separable and non-separable forms with different meanings. Wikipedia on separable verbs has the verb "umfahren" as example:
contiguous: Ich umfahre den Fußgänger. I (am) drive (driving) around the pedestrian.
separated: Ich fahre den Fußgänger um. I (am) drive (driving) over the pedestrian.
In perfect, the prefix isn't separated but in the "separated" form, there's an addition prefix "-ge-":
contiguous: Er hat den Fußgänger umfahren. He has driven around the pedestrian.
separated with additional prefix: Er hat den Fußgänger umgefahren. He has driven over the pedestrian.
contiguous: Wir umfuhren den Fußgänger. We drove around the pedestrian.
separated: Wir fuhren den Fußgänger um. We drove over the pedestrian.
contiguous: Er hat versucht, den Fußgänger zu umfahren. – He had tried to drive around the pedestrian.
separated (only by "-zu-"): Er hat versucht, den Fußgänger umzufahren. He had tried to drive over the pedestrian.