People often grudgingly say this, failing to complete the quote.

They usually mean ‘You should look after your own first’ which is not the original meaning of the saying!
The actual meaning is that teaching charity begins in the home – children begin learning how to be charitable in their home environment.

Charity comes from the Latin word, caritas which means love.
Often attributed to 17th century writer Thomas Fuller, the saying is actually a centuries old observation that learning how to care for humankind at home encourages us to apply it elsewhere.

Rwandan ethos is an uplifting example. Even in the poorest of families a last bowl of rice will be shared with neighbours, if they too are starving. Strangers are welcomed with kindness and if you fall, a dozen people will rush to help you up.
On the last Saturday of each month, communities gather together from 7:00am to noon, working in unity to help each other. They repair buildings for the elderly or build schoolrooms for their children. They share goods with the needy or decide how to solve problems in their village.
It’s called Umuganda – a Rwandan tradition that dates back long before colonial times.

And it’s how families start teaching their children to love and care for all humankind.