Question: I'm currently working on a short story in which the main character has received a device from an anonymous individual. This device *appears* to function like a regular phone. Until he realizes that he can send text messages up to 1 hour back into the past. 

My question being... How would such a device (theoretically) function?

Note: Handwaving is allowed but do NOT make your entire answer one big handwave

Answer: 
Such a device is called a tachyonic antitelephone.
<https://en.wikipedia.org/wiki/Tachyonic_antitelephone>
It relies on the existence of tachyons - particles that always go faster than light. 

This Worldbuilding idea has a fine answer walking through why FTL implies time travel.
[Are there any ways to allow some form of FTL travel without allowing time travel?](https://worldbuilding.stackexchange.com/questions/46873/are-there-any-ways-to-allow-some-form-of-ftl-travel-without-allowing-time-travel)
It's not possible
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Imagine this:

A phone (let's call it phone A) gets destroyed. 30 minutes later, the *device* sends a message to phone A 1 hour in the past. There is *no* way that phone A could receive that message since it already got destroyed. It can no longer receive any message ever since it got destroyed, even if that message was sent before it got destroyed. This is the same problem with time travel. You cannot bring a dead person back to life in the real world.