Learn French: Je mange le reste demain. - I am eating the rest tomorrow.

Yes! You can say J'en mange le reste demain to mean exactly that. The pronoun en replaces de plus the thing you are eating, giving the meaning of 'of it'. However, as this sentence shows, just saying le reste is also perfectly fine when what you are eating is obvious from the context.

Why is the present tense mange used if the action happens demain (tomorrow)?

Just like in English, French often uses the present tense for future events that are planned or happening very soon. You could also use the near future tense and say Je vais manger le reste demain, but the present tense is very common and natural here.

Does le reste also mean 'rest' as in relaxing or taking a break?

No, le reste only means 'the rest' in the sense of a remainder or what is left over. If you want to talk about resting to get your energy back, you would use the noun le repos or the reflexive verb se reposer.

If I wanted to specifically say 'the leftovers', would I just make it plural?

Exactly. To talk about food leftovers, you use the plural form les restes. For example, Je mange les restes demain means I am eating the leftovers tomorrow.