What is the intuition behind exponentials in the frequency domain signifying delay in a signal?

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Question is in the title.

I am trying to teach myself control theory and it involves a lot of taking things as truth without much intuition or proof. In particular, I'm trying to understand how $e^{s \tau}$ represents some form of delay.

For context when you answer, please consider that I am only familiar with Laplace transforms as a technique to algebraically solve differential equations in a very abstract context.

Thank you