You are being confused because of the misleadingly nonstandard pronunciation symbols being inexcusably thrust upon you by Merriam-Webster. It it in your best interest to ignore utterly any source whose pronunciation fails to use standard IPA.
Turning to a dictionary that does employ standard notation, the related word cactaceous, meaning per the OED:
Belonging to the old genus Cactus; or to the natural order Cactaceæ.
is normally pronounced /kækˈteɪʃəs/ according to that same source.
That tells you how Cactaceae starts out, but leaves you a bit in the dark about how it finishes up. Most of the remaining mystery lies in how the ‑aceae suffix can be pronounced several ways, depending on one’s formative education and personal predilection.
So for example, although I myself — and I am hardly alone in this — pronounce the word Cactaceae as /kækˈteɪʃiɑɪ/, other people are more apt to say /kækˈteɪʃieɪ/ or /kækˈteɪʃeɪ/ there instead.
I have never heard /kækˈteɪʃi/ but can imagine it existing. Probably very few still pronounce it as if it were Classical Latin, so /kɑkˈtɑkiɑɪ/ or even with some small effort to preserve the diphthong, /kɑkˈtɑkiɑe/.
Mostly what is happening is that the /si/ becomes (as it so often does) /ʃi/ there; think of spacious and many other such words.
I recommend to you the Wikipedia article on the ‘traditional’ English pronunciation of Latin. This is still true in general, but studious specialists of a younger generation sometimes do more in the way of maintaining the Latin diphthongs than our elders were wont to do.
As John Lawler astutely notes in comments, it may be important to preserve an unpalatalized /s/ instead of using /ʃ/ for the ‑aceae suffix in some contexts, because that suffix is a clear marker to botanists that one is referring to a plant taxon at the Family level.
That would make it end in /esie/ phonemically — or, with explicit phonetic diphthongs and glides as always occurs in English, in [eɪsijeɪ]. It really depends on how fancy you are getting with your notation, because those are the same thing.
The Anglophone botanists I know indeed use a simple /e/ (phonetic [eɪ]) for ae. I may hear the [ɑɪ] variant mainly from people who have spent more time with school-Latin than with actual English-speaking botanists. :)