The two sentences have a different meanings.

Je les ai achetés.

Indicates that you buy something (les), so it could be translated like "I’ve bought them"

Je leur ai acheté.

Indicates that you buy something from somebody (leur), so it could be translated like "I’ve bought from them."

• J’ai acheté les chemises → je les ai achetées.

• J’ai acheté les livres → je les ai achetés.

• J’ai acheté quelque chose à quelqu’un → I bought something from somebody/ I bought something for somebody.

• Je (les) leur ai acheté → I bought (them) from them/I bought (them) for them.

EDIT (merci @Xoudo)
One must mention "Je les leur ai acheté" is quasi never used. "Je le lui ai acheté" is much more common.