Learn Swedish: Vill du kasta det avklippta håret? - Do you want to throw away the cut-off hair?

This is the rule of obligatory prefixing! When a phrasal verb like klippa av (to cut off) is turned into a past participle to be used as an adjective, the particle (av) must be attached to the front of the word as a prefix. You cannot say det klippta av håret; it must become det avklippta håret.

Why does avklippta end with an -a in this sentence?

Because it is acting as an adjective in a definite phrase. The base past participle is avklippt. However, since the noun phrase is definite (det ... håret), the adjective must take the definite -a ending, becoming avklippta.

The English translation says throw away, but the Swedish sentence just uses kasta. Where is the word for away?

While kasta literally translates to throw, in everyday Swedish it is very commonly used on its own to mean throw away or throw in the trash. You could explicitly say kasta bort or use the verb slänga, but using just kasta is perfectly natural in this context.

Just to review, why do we need both det and the -et ending on håret?