"I am at home."
it could refer to any house, but most people will think of their own home unless previous conversation indicates something else. I would always say "evdeyim" when I am at my own home, "Evimdeyim" is grammatically correct but sounds odd, I would only use it if someone asks me in which house I am.
To make it a bit clearer, the first -im suffix is the genitive turning the word into 'my house', and the second one is the personal ending which turns the word into a whole sentence making it behave like a verb.
Imagine saying "Evimdesin.", which means "You are at my house." while "Evimdeyim." means "I am at [my house] home.".
There is no such word as "evideyim" in Turkish, there are some combining letters (this one seems to be 'n' of -y-, -ş-, -s-, -n- set) we use to ease the language usage.
I do not really know or remember why that 'n' is used in "[Onun] evindeyim." meaning "I am at his place." as the word does not violate any rules like "No two vowels can be used next to each other without any combining letter in between.", but it is there for some reason...