[v1: drop る] + させてください [v5: う-row → あ-row] + せてください [する → させてください] [来る → 来させてください]
させてください means "please let me": you ask permission to do something yourself. It is the causative て-form plus ください. A る-verb drops る and adds させてください; a う-verb moves its last sound to the あ-row and adds せてください.
A う-verb takes せ, never させ.
At work and in formal speech, させてください also introduces something you are about to do anyway. Asking permission makes it polite.
させてください is polite. To a customer or a superior, させていただけませんか is more polite still, and させていただきます announces your action without asking. Among friends, させて on its own does the job.