A-type blood has B-antibodies; it also clumps anti-A antibodies are inserted. Why is that? The blood has no antibodies against the A-antibodies to make clump in this way.
When an antibody meets its antigen it will bind it. So if you add anti-A antibodies to A blood, these antibodies will bind the red blood cells. Because of its Y-shaped form, each antibody can bind two epitopes. These can be located on the same red blood cell, or on two different. If they are located on two different, this leads to cross-linking of two blood cells. Since this happens many times, the blood cells will clump. The whole process is called hemagglutination.
See the image (from here) for illustration:
The so-called ABO blood grouping is based on the antigen-antibody reaction, where anti-A antibodies circulating in the plasma will bind to cells possessing antigen-A and cause clumping, or agglutination. A deadly situation since the red blood cells will be unable to circulate.
A person's blood group is designated according to the antigens found on the surface of their red blood cells. Type A blood means type A antigens are on the surface of the red blood cells. Type B - the same. Type AB - the same. Type O blood means there are no type-A or type-B antigens on the red blood cell's cell membrane/surface, theoretically making them the universal donor of red blood cells, but not their plasma if it contains anti-A, or anti-B, or both anti-A and anti-B antibodies, that can bind to their pair antigen type.
Persons of type-A blood mean that their red blood cells exhibit the antigen-A on their red blood cell membranes/surfaces, hence these persons can safely have anti-B antibodies circulating in their plasma to avoid agglutination. Type-B blood therefore means that the plasma may have anti-A antibodies safely.
Blood matching is further complicated by various blood antigen systems that have been found to be clinically significant, for example the Landsteiner blood antigens.