When blood is donated, the antibodies within it are extracted, but how exactly do they do it? How do they take out the antibodies within the blood, what process do they go through?
I don't believe they do remove the antibodies but rather they match and screen the donor antigen and antibody profiles to those of the recipient. The process is very briefly outlined on the Red Cross website;
Most blood is spun in centrifuges to separate the transfusable components – red cells, platelets, and plasma
The primary components like plasma, can be further manufactured into components such as cryoprecipitate
Red cells are then leuko-reduced
Single donor platelets are leukoreduced and bacterially tested.
The article in wikipedia explains the screening for blood transfusions;
Patients should ideally receive their own blood or type-specific blood products to minimize the chance of a transfusion reaction. Risks can be further reduced by cross-matching blood, but this may be skipped when blood is required for an emergency. Cross-matching involves mixing a sample of the recipient's serum with a sample of the donor's red blood cells and checking if the mixture agglutinates, or forms clumps. If agglutination is not obvious by direct vision, blood bank technicians usually check for agglutination with a microscope. If agglutination occurs, that particular donor's blood cannot be transfused to that particular recipient.