In real life, you would either depend on a third party API to fetch the colored coins of a transaction or a balance. Which might not be a good idea, because it add a trust dependency to your program with the API provider.

When using either you need to have an or counterpart.
Example: You can either go left or right.

The author of the quoted text might be trying to tell us that,
in order to fetch the coloured coins of a transaction, we can
either use a third party API or a balance.

But I think it is more likely that the or part of the sentence was simply omitted.

NBitcoin allows you either to depend on a web service, either to provide your own implementation for fetching the color of a Transaction. This allows you to have a flexible way to unit test your code, use another implementation or your own.

I think the author means allows you to either depend on a webserver or provide your own implementation to fetch the color of a transaction