Technically it's the "lagar" that means "cooks" here, not "lagar mat" as a whole. The difference is that unlike in English, "lagar" requires an object, you can't say just "jag lagar" like you can say just "I cook" in English.
However, you might not feel the need to specify what exact dish someone is cooking. In this case you use "mat" or "food" as sort of a dummy object. What are you cooking? You're cooking food, you don't need to specify the exact dish. Hence, "lagar mat" can be translated as "cooks food".
However, as I earlier mentioned the English verb "cook" doesn't require an object, and translating "laga mat" overly literally as "cook food" would sound unnatural. That's why you simply ignore the "mat" in the Swedish phrase, which is only there to fill the place of an object that the English equivalent word doesn't mean.
In the sentence "Jag lagar min egen mat.", "min egen mat" ("my own food") is the object, so there's no need to use a lone "mat" as a dummy object.
It is an adjective. Or so SAOL tells me. (http://www.svenskaakademien.se/svenska_spraket/svenska_akademiens_ordlista/saol_13_pa_natet/ordlista)
It's an odd word for sure and it isn't really intuitive to think of it as an adjective. I don't know how they usually classify own in English, but the first search hit I got puts it as an adjective too (as in my own book) so it may be a traditional thing to classify it that way.