Hey! There is nothing to worry!
Sin/sit/sine are only used for the third person singular cases (He/She/The girl/The boy etc etc) and there is a logic to why it is used. It is just the language's way of providing more information! For eg:
"Han læser sin avis" equates to " He is reading HIS newspaper". It shows that he is reading his own newspaper and not someone else's. Similarly, in this example the girl is eating her own food not her friends' or anyone else's. The language is just trying to make that distinction.
The only thing you've to know here is whether the object is a common or neuter word. That'll come with practice.
Many languages have gendered nouns. "Gender" is just a rather arbitrary word for splitting the nouns up in two or more categories. "Mad" happens to be the common gender or "n", and "brød" happens to be the neuter gender or "t". You'll have to learn which is which by rote. The difference the gender makes is in whether words associated with it tends to get n's er t's as suffix.
In Spanish and French the genders are actually called "male" and "female". That does not mean that houses in Spain are actually female(!) It just means that "the house" in Spanish is "la casa" not "el casa".
"Sin/sit/sine" always mean that the subject of the sentence owns the object. In this example "pigen" is the subject, and as "pigen" is a female human, the possessive pronoun in English that must be used is "her". If you were to say "Pigen spiser hendes mad" then it would mean that the girl is eating another female's food.
If the subject of the sentence is a male human then the possessive pronoun "sin/sit/sine" translates to would be "his".
Drengen spiser sin mad = The boy eats his (own) food
Drengen spiser hans mad = The boy eats his (someone else's) food
If the subject of the sentence is an inanimate object, an animal (though this depends on the speaker's relationship to the animal), then "sin/sit/sine" translates to "its".
Hopefully that makes a bit of sense and has cleared it up for you.