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I was wondering, what the etymology of Latin infinitive verb endings -are, -ere and -ire was. I assume they are Indo-European, but I haven't found any information about it.

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The Latin infinitive verb endings -are, -ere and -ire can be analyzed as the thematic vowels of the verbs -a-, -e- and -i- plus -re < *-s-e(y), with rhotacism r < s. The same morpheme *-s- exists in Greek -ein < *e-s-en with intervocalic loss of s. You should find more detailed information in manuals.