This phrase has a dangerous insulting subtext, which can cause the speaker trouble. It means that something is, but was unexpected. The subtext adds a second meaning. The phrase can be used innocently, for example:
Welcome to our board meeting everybody, today happens to be the beginning of a new fashion season.
Our new TV happens to be a web-browser as well.
This chemical we are using happens to be very poisonous.
In these three examples there is no insulting subtext. The meeting is on the day a new season's clothes start being sold. The TV has modern internet-capable functions. Unfortunately the chemical you need is poisonous; be careful!
The phrase can also be used to refer to a person, but that is when the dangerous subtext begins.
Mr Chan, our strict CFO, happens to be a grower of prize-winning flowers.
The winner of the kung-fu tournament happens to be a kindergarten teacher.
Mr Chan is a tough, ruthless businessman; but there is no reason he cannot be good with flowers, it just seems unexpected. A kung-fu champion is likely to be an emotionally calm person; however gentleness is just not the impression one gets from a person who can break masonry blocks with her fists. These examples are not too insulting.
The trouble is that this phrase gets very insulting when it is used to describe minorities who break established stereotypes.
The doctor who fixed my eye happens to be African-American.
The electrical engineer happens to be a woman.
There is no reason why an African-American person cannot be a doctor or a woman cannot be an engineer. To avoid misunderstandings, it is best not to refer to sex, race or ethnicity, unless it is relevant. For example: The patient will want the dark brown artificial leg to match his skin colour. The phrase happens to be is used for a subtle sort of racism. Nothing bad is said directly, but it implies that a situation is abnormal.