Learn Icelandic: Ég les bókina heima. - I am reading the book at home.

The dictionary form bók just means "a book". Because we want to say "the book", we have to add a definite ending to the word. Furthermore, because the book is the direct object receiving the action of the verb les (read), it must be in the accusative case.

How do we know to use -ina as the ending for "the book"?

bók is a feminine noun. In the accusative case, the definite ending (meaning "the") for feminine singular nouns is -ina. If it were the subject of the sentence (nominative case), it would just be bókin.

Does les translate exactly to "am reading"?

Yes, in this context! Icelandic does not have a separate "-ing" tense (continuous tense) like English does. Ég les can mean "I read" (every day) or "I am reading" (right now).

Why is the word for "at home" heima instead of just heim?

In Icelandic, words for locations often change depending on whether there is movement. You use heima when you are stationary or located "at home". You would use heim if you were traveling "to home" (in motion). Since you are staying in one place reading, heima is correct.