You asked two things:
- I eat cheese, why don't we use I?
- I cook the chicken, why do we use the?
1 : I eat cheese, why don't we use 'I'?
The thing is that in italian we can omit the subject because any predicate has its own different declinations for each of the basical subjects. This means that if you say
Mangio il formaggio
Mangio is used just as first singular person, so it always means I eat.
The reason why you always use I is because the predicate is basically the same for first and second singular and plural persons. Indeed the sentence eat cheese can be used as it is, in english, by the following subjects:
I, you, we, you (multiple people), they
In italian things are different, predicates have their own subject implicitly exposed their declination
Mangio il formaggio | (Hidden I: Io)
Mangi il formaggio | (Hidden You: Tu)
Mangia il formaggio | (Hidden He/She: Egli, Ella)
Mangiamo il formaggio | (Hidden We: Noi)
Mangiate il formaggio | (Hidden You (pl): Voi)
Mangiano il formaggio | (Hidden They: Essi, Loro)
Please have a look here for further informations about italian regular predicates usage:
2 : I cook the chicken, why do we use 'the'?
Because in italian we pretty much always use articles.
There are two types of articles, like in english: definite article, and indefinite article.
Definite articles are
Il, lo, la, i, gli, le
Undefinite articles are
Un, uno, una
The usage is very close to the english but it has an addition: they have to be ever used with common nouns or pronouns. The only exception is with proper nouns, which basically don't need the article (even if in some italian regions the dialect involves the use of it in this way).
Let's go over some example:
Grandmom cooks chicken:
La nonna cucina il pollo
Kim cooks chicken:
Kim cucina il pollo
The exception involves just the proper nouns or pronouns, let's say that common nouns can't be used without articles.
Of course this also means that our definite articles aren't so tight as the english ones. Saying
the chicken in english means
that specific chicken, while in italian doesn't, even if it could.
Another great example involves the possessive adjectives, here you don't put any kind of article, while we do: indeed
my car becomes
la mia macchina
About italian possessive adjectives: