Learn Russian: Я работаю дома.

This shows the action belongs to Я (I). The dictionary form of the verb is работать (to work). For most verbs ending in -ать (known as First Conjugation verbs), you create the "I" form by dropping the -ть and adding -ю.

How would I change this to say "you work at home"?

To say "you work" using the informal "you" (ты), you take that same dictionary form работать, drop the -ть, and add -ешь. The sentence becomes Ты работаешь дома.

Why is the word for home дома instead of just дом?

The word дом is a noun meaning "a house" or "a building." However, дома is a special adverb that specifically means "at home" or "in the house." You use дома whenever you are talking about the location where someone is or where an action is happening.

Why is there no Russian word for "at" in this sentence?

In English, we need the preposition "at" to say "at home." In Russian, the single word дома has the "at" meaning built right into it, so no extra preposition like в (in) or на (on/at) is needed.