What did Yakov mean when he instructed the messengers to tell Esav that he was following them?
Ramban: Refer to 32:21:2:3.
Hadar Zekenim: Hashem sent the angel to stop him from fleeing. This was a punishment for fleeing Lavan, and not trusting in Hashem's promise. Rashbam (25) - it was in order to show Yakov the fulfillment of Hashem's promise, that Esav will not harm him.
This is why he crossed over the River Yabok at night with his entire family (in Pasuk 23), like David ha'Melech, who crossed the River Yarden in order to escape from Avshalom (Rashbam).
What was Yakov trying to achieve by sending Esav this gift? What does "Achaprah Fanav" mean?
Targum Yonasan: His plan was to appease him before meeting him face to face.
Ramban: After informing Esav that his servant Yakov was following them, he instructed his messengers to tell Esav that he was sending his honor 'a redemption of the soul' (Kofer Nafsho), in the way that servants enter the presence of the king with gifts, after he has granted them the privilege of a private audience...
כבר אמרו לעיל "גם הוא [-יעקב] אחרינו"?
גור אריה (לעיל, י"ט): כפלו כאן להוסיף שיעקב אמר "אכפרה פניו" לבקש את מחילתו.
מי אמר "כי אמר אכפרה פניו"?
אבן עזרא: יעקב אמר כך בליבו ואלו דברי משה.
גור אריה (לעיל, י"ט): השלוחים אמרו כך לעשיו בשם יעקב.