In Bitcoin blockchain, Segwit takes the script, which containers the signature and sender's public key, out of each transaction and stores them into an extended block. However, why having an extended block can make the blockchain faster? Since I think each miner now has to process the original block plus the extended block, thus the amount of data each miner needs to be processed has not been changed. Please advise!
First of all, your understanding of segwit is incorrect. There is not extended block, the witness data is still part of the transaction.
Segwit also does not "make the blockchain faster". What part of it does is it redefines the maximum block size metric to be something called block weight. Witness data is weighted differently from non-witness data, which means that a block containing witness data will have more spare weight units to contain more transactions. This means that Bitcoin's transaction capacity will increase, as will the size of blocks.