Learn Farsi: گربه‌ی این مرد گرسنه است.

This is the Ezafe particle showing possession, linking "cat" to "man" (literally: the cat of this man). Because گربه ends in a silent 'heh' (ه) which makes an "e" vowel sound, we must use ی (-ye) instead of the standard invisible "-e" sound to connect the words smoothly.

In English we say "this man's cat". Why is the order changed to گربه‌ی این مرد in Farsi?

In Farsi possession, the item being possessed always comes first, followed by the Ezafe, and then the owner. So we literally say "cat-of this man". The word این (this) stays right next to مرد (man) because it belongs to him, not the cat.

Why isn't there an Ezafe linking مرد (man) and گرسنه (hungry)?

Because "hungry" isn't an adjective directly describing "man" here. Instead, it's the state of the cat! The sentence structure is "[The cat of this man] [hungry] [is]". If you wanted to say "this hungry man", you would use an Ezafe: این مردِ گرسنه.

Both گربه and گرسنه end in the letter 'heh' (ه). Do I pronounce it as an "h"?

No. At the end of these specific words, the 'heh' (ه) is silent and instead acts as a short "e" vowel (like the 'e' in "bed"). They are pronounced "gorbe" and "gorosne". This vowel sound is exactly why گربه needs the special ی (-ye) form of the Ezafe!