I can't find any explanation of that diacritic anywhere, neither in the IPA chart nor at Wikipedia's articles.
The diacritic ̯ means "non-syllabic", and it typically used to refer to phonetic situations where a vowel seems to be pronounced as "a glide". You could write the IPA letter [j] as [i̯], but there is a special symbol for non-syllable /i/. There is no special symbol for non-syllabic [e], hence the need to resort to the diacritic in [̯e]. The reason for the lack of a special letter for non-syllabic /e/ is that the association hasn't approved any such letter, which is because it has not been proposed, which is because it is not phonemically distinctive in any language (or, so it seems).
The combining breve below means "non-syllabic" - to contrast with the normal idea of vowels as syllabic (form the nucleus of a syllable). Modern linguists tend to notate parts of a diphthong with a non-syllabic consonant like over an equivalent semivowel like in diphthongs because:
1) some languages do have diphthongs with a semivowel that does not occur as an independent consonant. One example is German, with /aʊ̯/ as a diphthong, but no /w/ as a consonant, even though /ʊ̯/ and /w/ mean about the same thing.
2) It's rare to see a semivowel that occurs in diphthongs if it doesn't occur as an independent vowel as well. If a language has /ɥ~ʏ̯/ (a semivowel that can be notated as either ɥ or ʏ̯), then it is very unlikely to not have the corresponding vowel /y/
3) In some languages, diphthongs act more like vowels than a vowel+consonant sequence, and so having notation that doesn't look like you are claiming there is a consonant in the diphthong is useful.