Humans have a number of different blood groups. The two most people have heard of are the ABO and Rhesus blood groups.
The negative (–) and positive (+) in your question refer to the Rhesus blood group. In my answer I shall ignore it because it unnecessarily complicates matters.
The letters refer to the ABO blood group system, which, with the Rhesus blood group, is one of the two that are usually typed when blood is grouped. In the ABO blood group system a person can have one of four blood groups (phenotypes). They are A, B, AB and O. Of those four we know in the case of two what the person's genotype (alleles for the ABO blood group) is. People who phenotype is blood group AB have the alleles A and B. If someone has the ABO blood group (phenotype) O they have two O alleles.
With the above information in mind we know the AB father has alleles A and B whilst the O mother has two O alleles. If the man has any biological children he will give some an A allele and some a B allele. This woman would only be able to give her offspring an O allele. As a result children who had this man and woman as their biological parents could have one of two possible genotypes, i.e. AO or BO. Their resulting ABO blood group (their phenotype) would be A or B, respectively. A man with the ABO blood group AB and a woman with ABO blood group O cannot have a child with ABO blood group AB. As you can see this was resolved by reference only to the ABO blood group system.
As a quick aside, if at least one of the parents can give an allele for Rhesus +ve the child can be Rhesus +ve.