Learn Russian: Я болен, поэтому я дома.

болен is a "short-form" adjective, which Russian often uses for temporary states like being sick. It has to match the gender of the person it describes. болен is masculine, so the speaker here is a man. A female speaker would say Я больна.

Where is the word for "am" in this sentence?

In Russian, the present tense of the verb "to be" (am, is, are) is usually left out. You just put the subject and the description together. So, the sentence literally translates as "I ill, so I at home."

I often mix up поэтому and потому что. Why is it поэтому here?

поэтому introduces the result (so, therefore, that is why), while потому что introduces the reason (because). Since staying home is the result of being sick, you use поэтому. If you reversed the sentence, you would use the other one: Я дома, потому что я болен (I am at home because I am ill).

Why is "at home" just дома instead of using a preposition like в (in)?

дома is a special adverb that simply means "at home." You do not need a preposition. If you said в доме, it would literally mean "inside the house" (referring to the physical building), which sounds unnatural when you just mean you are staying home.