my learning of this one took some time... so based on my personal experiences so far, this is what I know...
" मैं समझती हूँ " is the simple verb form (and masc/fem endings). It's not as common in everyday speech, and it's more formal.
I translate it literally as "I DO understand". do goes with the हूँ "hun" ending.
"मुझे समझ आया" is a more commonly used form, and I translate it as "I've COME to understand" aka "i've got it".
aaya is there at the end, meaning "coming". I've heard various tenses used when "aana" is at the end of the root verb "samjh".
"मैं समझ गयी" is also used frequently... and with my urdu and hindi speaking friends "gaya", meaning "went", is used quite a bit with other verbs, or perhaps I've picked up on it more since "gaya" has been difficult to understand! Anyway...
literally, I interpret this phrase as "I WENT understanding" aka "I've understood". "gayi" for female, "gaya" for male.
Then using "समझी" is just the root of "understand" by itself, which is fair game for the different verb tenses. Also very commonly used in slang speech. You can say: "samjhey?" to ask someone, "understand?"
and I could reply, "main nahin samjhi"... as in, "I don't understand".
and for boys they could say "main samjha" and "main nahin samjha" for "I understand/ I don't understand".
Again, हूँ "hun" is not there at the end, so it's not literally "I do understand" but only just "I understand". "do" is rather invisible I guess, haha.
I hope this helps you some! Re-explaining it has helped me. and I welcome corrections if I've spoken incorrectly. :)