How do animals avoid the problem of inbreeding?
The question is interesting. However, it will be impossible to answer accurately for all birds as the mechanisms actually used vary from species to species. So I will give you some information in the general sense.
Recurrent inbreeding reduces inbreeding depression
Inbreeding depression is caused by both

Recessive deleterious alleles that are brought to homozygosity
Some epistatic interaction

In any case, in a population where inbreeding is common, those recessive deleterious alleles and alleles involves in these deleterious epistatic combinations will be washed out (or at least brought to lower frequency) of the population via selection and the population will end up suffering less from inbreeding than a typical outbreeding population. In the extreme, a population of exclusively selfer individuals, such deleterious recessive alleles are brought to extremely low frequency within lineages.
How to avoid inbreeding?
Now, sure animals and plants typically don't have a big book that allow them to tell how related they are to another individual. However, there are other mechanisms of kin detection
Self-incompatibility
For example, in plants self-incompatibility is thought to be relatively uncommon. There exist many mechanism, so I-ll just consider a simple example. Consider a self-incompatibility locus as a specific loci for which if the same allele is shared by both the pollen and the ovule, then no fertilization occurs. This prevents selfing and prevents mating with many of the closely related mechanisms.
Body odour and sexual attraction
Also, there are mechanism that can mediate sexual attraction toward individuals that differ. The classical example is Major histocompatibility complex (MHC) and sexual attraction (vertebrates), where individuals tend to be attracted to individuals that have a different MHC.