How much transaction fees are taken by bitcore for API use? If I am using the API of bitcore.io and create transactions, how much transaction fees are taken by Bitcore?

I couldn't find any description in their documentation.
Bitcore is open source software provided at no charge, so there is no fee you have to pay to the BitCore software authors.

When creating a Bitcoin transaction, it is highly recommend that you pay a fee to Bitcoin miners to encourage them to include your transaction in a block.  How much fee you pay is optional, and there are a number of fee estimation services that will recommend a fee for you.  See also other questions on Bitcoin StackExchange for more information about transaction fees in general.