Learn English – Why are we supposed to say the “a” as an “e” in “any” and “many”

They're not all like [ɪ]. They're just different from other dialects. The "vowel" section of the Wikipedia article about New Zealand English discusses how New Zealand vowels are different from other dialects.

Note, however, that /ɪ/ itself is pronunced [ʌ], that is, "sin" sounds almost like how I would say "sun".

As for why, nobody quite knows where sound changes originate. In many languages, including various times in the history of English, when one vowel changes, many others change as well, to redistribute them throughout the vowel space. Such groups of changes are known as chain shifts, the most famous being the Great Vowel Shift. This is clearly what has happened with short vowels New Zealand English. The description in the Wikipedia article says "The short-e /ɛ/ of YES has moved to fill in the space left by /ɪ/, and it is phonetically in the region of [ɪ]", implying this is a pull-chain—a shift of one vowel opens up room in the vowel space that other vowels move into.

I don't know if there are any studies that show the history of the New Zealand vowel shift, so I can't be sure the implication from Wikipedia is correct, as it is uncited, but it is certainly a plausible explanation.

A very reliable rule of thumb for British English and General American is this:

• If the syllable con is stressed it will take a full strong, vowel, and the first syllable will be: /kɒn/.

• If the syllable is not stressed we will find a schwa, /ə/, or no vowel at all in the first syllable: either /kn/ or /kən/.

However, we should bear in mind that some regional varieties of English will have a full vowel in the first syllable of many of these words regardless of stress. So in Yorkshire English, for example, many speakers will say /kɒnˈstɪtʃʊənt/.

This morpheme 'con' occurs a lot in English. That last sound in 'con', /n/, is very unstable both because it is an alveolar sound and also because it is nasal. It tends to change according to the following consonant. We find this prefix with different final consonants in words like: collect, commemorate, correlate. These types of word also show the same variation in their first syllables depending on the stress:

• colleague: /ˈkɒli:g/
• collection: /kəˈlekʃn/
• compensate: /'kɒmpənseɪt/
• computer: /km'pju:tə/
• correlate: /ˈkɒrəleɪt/
• corroborate: /kə'rɒbəreɪt/

Both words collection and corroborate could be said with no schwa at all, and just a syllabic consonant - but the transcription for this is tricky in terms of syllabification, so I have taken the easy route and used a schwa. Similarly, computer could equally be said with a schwa instead of a syllabic /m/.

[Transcription note: in line with the Original Poster's question I have used Southern Standard British English transcriptions].