Translation:The river behind the city is wide and deep.
Yes, that's how it works. The main role of the suffix -i is to turn nouns that refer to a location into an adjective. Amerika -> amerikai (American / of America); Budapest -> budapesti (of Budapest).
This doesn't work with all such nouns, and, as you can see, not only nouns can take it.
I know there is someone who is spreading this i stuff but just leave it... it is true that in many languages (including Hungarian) /j/ could be seen as a very short i but we do have j as a separate letter so there isn't much point in it. "mély" is pronounced just like "méj" would and that's that.
Now, having said that, I understand why you bring up í instead of é, for this given pronunciation. It could be better articulated, if you ask me. However, to some extent, the phenomenon is legit - as I said, j works similarly to a very short i so it's somewhat natural someone shapes their mouth as if they were to pronounce an i during the span of j. Therefore, that é might shift a bit towards í. Here, I think it's a bit like a long version of the "i" in "this" - more open than the Hungarian "i" but more closed than "é". Also, there should be a noticable difference between (I'm only writing hypothetical words with Hungarian phonetics now) "mé" and "méj", also between "mí" and "míj". You will figure this out when you hear the Hungarian pronunciation of "okay", ie "oké". A closer approximation of the English version would be "okéj".