I still got the impression that you did not really have read my answer to a related question.
But still, I try to briefly answer your questions here. First of all, private key extraction essentially means private key generation. Extraction, because the partial private key ($D_A$) is generated with repsect to an identity string $ID_A$ uniquely identifying the user $A$ (and thus it may be viewed as extracting the private key from the string).
Why is there a need for a "partial private key extract algorithm"?
Because the certificateless-approach is in the middle between traditional public key crypto (key pairs are generated by the users themselves) and identity based cryptography (private keys are generated for the users by a trusted authority - this is necessary, since public keys are identity strings and otherwise everybody could generate private keys for any public key - and this would be quite bad).
Consequently, in public key crypto, public keys need to be certified by another party and in identity based crypto they do not need to be certified, since a user has to prove the identity to the trusted authority before the private key is generated.
In certificateless cryptography a part of the user private key is generated by a trusted authority (partial private key extraction) and the other part is computed by the user locally.
Why should it be needed when calculating "user's private keys by user"?
It is needed since users do not compute their entire private keys on their own. Then, they would need to certify their respective public keys (who will certify that they are indeed who they are pretending to be in certificateless crypto?). So a partial private key is issued to users after identity verification by the trusted authority. The remainder of their private key is chosen on their own and used to compute the public key (without requiring to certify it).
Is "partial private key extract algorithm" further used in other algorithms?
No, this is only used for key generation (of the partial private keys of users).
It is important to understand that in certificateless encryption, a message is encrypted with respect to the user's identity ($ID_A$ - which is used for by the trusted authority to generate the partial private key) and the public key $P_A$ (this one does not need to be certified). The ciphertext can then be decrypted using the knowledge of the partial private key $D_A$ and the second part of the private key $x_A$ only known to the user.
Someone that encrypts a message with respect to $ID_A$ and $P_A$ can be sure that only the owner of the private key corresponidng to $ID_A$ can decrypt (this is the only user who will get issued the partial private key after identity verification at the trusted authority) and without the requirement that $P_A$ needs to be authentic (thats the idea).
Sorry, if there are many repetitions in the answer, but I tried to make the essential poits as clear as possible ;)