Learn German: Ich liebe dich.

dich is the accusative form of du (you). We use the accusative case for the direct object of a sentence, which is the person or thing directly receiving the action. Since "you" are the one being loved, the pronoun must change from du to dich.

If dich is "you" as an object, how do I say "me"? How would I say "You love me"?

To say "me" as a direct object, you use the accusative form of ich, which is mich. "You love me" would be Du liebst mich. Notice how du is now the subject doing the loving, and mich is the one receiving it!

Could I use the formal "you" (Sie) in this sentence?

Grammatically, yes. The accusative form of the formal Sie is exactly the same: Sie. So you could say Ich liebe Sie. However, in real life, it is extremely rare to declare your love to someone you aren't close enough with to use the casual du!

Just a quick refresher: why does the verb end in -e instead of -en?