All it does is to emphasise that someone has gone about something in a very businesslike way, with determination.
It is neither polite nor impolite. And it is not a radical expression. But it is often spoken with irony e.g. emphasising a business-like way of dealing with something that is not normally a business matter, in order to get over the idea of a determined approach. e.g. 'They made it their business to make me feel (un)welcome.'
Here are a few further typical expressions:
'I shall make it my business to help the new family settle in as smoothly as possible'.
' I shall make it my business to ensure that his request is turned down'.
'I shall make it my business to find out as much as possible about that company'.
'He made it his business to be as offensive as possible'
'She made it her business to be as helpful as possible'.