The direct Coombs test is easier to explain, so we can start there. I'll cover the indirect Coombs test in the next post.
Although there are other preparatory sequences, the main step in a direct Coombs test is to mix the patient's blood with so-called "antihuman antibodies". (Perhaps you're wondering how we manage to produce antibodies against our entire species? Naturally enough, we find a member of another species and inject the poor thing with our serum. It dutifully produces antibodies against our serum's constituents.) The theory is that these antihuman antibodies should bind to our antibodies.
In autoimmune haemolysis, this can produce an interesting affect, since this condition is already characterised by our own (human!) antibodies binding to our red cells. When we add antihuman antibodies to this mixture, these (rabbit/pig/yeti/whatever) antibodies bind to our human antibodies. But, since each IgG or IgM molecule has more than one binding site, each of these antihuman antibodies binds to more than one of our antibodies.
And, since our antibodies are stuck on to our red cells in autoimmune anaemias, this means that each of the antihuman antibodies effectively binds to more than one red cell at a time. (Are you keeping up?) The result, to the naked eye, is agglutination of the red cells. Although the test shown below is completely different, you can clearly see aggultination of red cells in the samples maked "Anti-A" and "Anti-D", when compared to the sample in the centre.
In summary, if the blood coagulates, you can infer that the patient's red blood cells have been bound by (his own) immunoglobulins. Of course, this isn't the normal state of affairs, and implies that the patient is experiencing an autoimmune haemolysis of his red cells. (Antibodies activate complement, and stimulate phagocytosis - neither of which is appropriate when they are attached to an erythrocyte.)
Fortunately, this has all been nicely depicted by at least one individual (click on it to see a bigger version):