させてもらう

させてもらう says you were allowed to do something and are grateful for it. It is the causative て-form plus もらう: よむ読む → よませて読ませてもらう, たべる食べる → たべさせて食べさせてもらう, する → させてもらう. The person who gave permission takes に.

させてもらう or てもらう

てもらう means someone else does something for you. With the causative, you are the one who does it, and the other person lets you.

Asking permission

As a question in the potential, させてもらえますか or させてもらえませんか asks to be allowed. The negative question is softer.

Announcing what you will do

In the non-past, させてもらう can announce something you have already decided, framed as if you had permission. It is polite on the surface and firm underneath.

させていただく

させていただく is the humble form, used to customers and superiors, at work and in speeches. Used for every action in casual talk, it sounds stiff.