Born to a very humble family?
I would say that the meaning here is "low in rank, importance, status, quality, etc.; lowly"
In this context, I would say that "very humble" is likely to be a euphemism for "poor". "Humble" could also mean "undistinguished" without being poor, but when combined with "very", it suggests some degree of poverty. It's also possible, but unlikely, that it means that they were shy about being wealthy (like Warren Buffet).
In this context, I would normally assume that humble is used to mean "not very wealthy".