Why do Ancient Greek words have "εί" from PIE "e"?

Question

Why do Ancient Greek words have "εί" from PIE "e"?

Ancient Greek κείρω <- PIE *(s)ker-.

Answer 1

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This is part of the "first compensatory lengthening", a set of regular Greek sound changes involving the loss of PIE/Proto-Greek *y and *s.

In this case, the full PIE proto-form was *ker-y-ō, with the present-tense formant *-y-. The sound change in Greek is:

[e|i|u][r|n][y] > [e:|i:|u:][r|n]

That is, when a vowel e/i/u preceded a resonant r/n which itself preceded *y, then the *y was lost and the vowel was lengthened. (There may have been an intermediate stage in which the resonant became a palatalized geminate, but this is uncertain.)

The spelling ει, as LjL notes, doesn't actually represent a diphthong here, but the long vowel [e:]. Some cases of ει go back to real diphthongs ("genuine diphthongs"), but others stand for the result of compensatory lengthening ("spurious diphthongs"); similarly for ου. The reason ει and ου were used for both types of sounds is that the original diphthongs underwent monophthongization, so that e.g. [ei] became [e:].

The change above is part of a larger set of changes that also involve other vowels and resonants, as well as *s in place of *y. For example, a sequence like *-ory- underwent metathesis rather than lengthening, ending up as οιρ, e.g. *morya > μοῖρα. To make things more complex yet, different Greek dialects had different outcomes for some of these sequences, so that Attic κείρω corresponds to κήρω and κέρρω in other dialects.