Learn German: Du rasierst dir die Beine. - You are shaving your legs.

Because the sentence specifies exactly what is being shaved: die Beine (the legs). When a sentence has a direct object receiving the action, the reflexive pronoun shifts to the dative case (dir). If you just wanted to say You are shaving yourself without mentioning your legs, you would use the accusative: Du rasierst dich.

Why does the German sentence say die Beine (the legs) instead of deine Beine (your legs)?

In English, we say your legs. But in German, when doing an action to your own body parts, you usually use a dative reflexive pronoun like dir to show whose body it is. Because dir already tells us it is your body, you just use the definite article die (the) for the legs instead of a possessive word.

Does the reflexive pronoun dir always come right after the verb?

In a normal statement like this one, yes. Reflexive pronouns usually stick right next to the conjugated verb (rasierst).

Does the verb rasieren always need a reflexive pronoun?

Only when you are shaving yourself! If you are shaving someone else, like a customer, you do not use a reflexive pronoun. For example, Du rasierst den Mann means You are shaving the man.