The infinitive form shows a diphthong /seɪ/, while the typical pronunciation of the third-person singular "simple present" form has /ɛ/ as the nucleus /sɛz/. Wiktionary suggests that the pronunciation /seɪz/ of "he says" is "dialectal". Does this occurrence happen with other verbs ending in -/eɪ/ (to my knowledge, "she plays" would be /pleɪz/; "it stays" would be /steɪz/)? If yes, then why did /sɛz/ become the usual form but not /plɛz/? If not, then can we account for the difference in a regular phonological way?
"Say" is an irregular verb, and also very common. As Peter Shor mentioned, "says" used to be pronounced /seɪz/ and is still pronounced that way in a few dialects, however mainly it is pronounced /sɛz/. Perhaps the best explanation is that mentioned by John Lawler, namely that as a result of being included in many idioms and being a very frequent word, the pronunciation has become "naturally compressed", although the written form remains as-is, for the time being at least.
I would add that this seems to be an example of how English pronunciation is evolving. The British Library has an ongoing project to track these too: https://www.bbc.co.uk/news/uk-11640951