Learn Farsi: اینجا جای من است.

Yes! The base word is jâ, which just means place. In this lesson, you learn that adding the indefinite -i makes it jâyi (a place or somewhere). But here, because we are talking about a specific, defined place (my place), we do not use the indefinite ending.

Why is there a ye sound at the end of jâ making it jâ-ye?

That is the Ezafe connector linking the thing (jâ) to its owner (man). Normally the Ezafe is just an e sound, but when a word ends in a long vowel like the a in jâ, Farsi adds a y sound to make it flow smoothly, creating jâ-ye.

injâ usually translates to here. Why does the English translation say This is my place instead of Here is my place?

The word injâ is literally made of in (this) and jâ (place). While it is often used to mean here in English, in Farsi it functions perfectly as a noun meaning this place. The literal translation of the Farsi sentence is This place is my place, which translates smoothly to This is my place in English.