"I.e." stands simply for "that is," which written out fully in Latin is 'id est'. "I.e." is used in place of "in other words," or "it/that is." It specifies or makes more clear.
"E.g." means "for example" and comes from the Latin expression exempli gratia, "for the sake of an example," with the noun exemplum in the genitive (possessive case) to go with gratia in the ablative (prepositional case). "E.g." is used in expressions similar to "including," when you are not intending to list everything that is being discussed.
EXAMPLES
I.E. (Id Est)
I'm going to the place where I work best, i.e., the coffee shop.
[There is only one place that I am claiming is best for my work. By using "i.e.", I am telling you I am about to specify it.]
E.G. (Exempli Gratia)
At the places where I work well, e.g., Starbucks, I have none of the distractions I have at home.
[There are lots of coffee shops I like, but Starbucks is the only international one, so it's the only "example" that would work.]
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