This could be a very long answer, but I'll try to keep it brief.
The first thing to understand is what causes the anaemia. I'm going to refer to defective haemoglobin found in a sickle cell patient as HbS and normal haemoglobin as HbA. Under certain circumstances (low O2 concentrations) the HbS protein is prone to aggregating into long filaments. Here is an image of a lysed erythrocyte releasing some of these filaments.
The presence of these filaments in the erythrocytes causes them to become misshapen (sickle-shaped) and this in turn causes them to get filtered out for destruction in the spleen which is continuously monitoring for defective erythrocytes (which could cause blockages in the circulation). This causes the anaemia (specifically a haemolytic anaemia, due to erythrocyte destruction, and nothing to do with a lack of iron).
The appearance of the ability to form filaments is due to a single Glu>Val mutation in the two β subunits of the haemoglobin tetramer. The new amino acid can fit into a pre-existing hydrophobic pocket on the surface of another subunit and so the molecules can daisy-chain to form the filaments.
The normal HbA can also bind on to a growing filament (it has the binding pocket) but that terminates the filament (because HbA doesn't have the Val residue). In this way people with sickle cell trait are protected from the anaemia by their normal haemoglobin.