Translation:Our family is big.
So there is a concept in Japan: うち/そと。uchi are the people in your inner circle and soto is outside of it, on an emotional level. This can be singular or plural. So when you hear うち don't just think We or I. Think an emotional feeling of close to home. An example of this that will give some culture shock... In America we have "common courtesy" extended to strangers and phrases like "we often hurt those closest to us". The Japanese will conversely give no attention to a stranger, ( そと) and save all their good emotional energy for those closest to them (うち). So the next time a Japanese person ignores you when you say excuse me, don't assume they're rude. Assume they're saving emotional energy and you are そと. Incidentally, white people will experience this the least because Japanese people are fascinated by them.
Maybe wn shouldn't think of them as seperate homonymn words, but one simpler idea that changes in meaning depending on the surrounding words. That word being close (emotionally) as someone else saying.
うち the noun being home, or place you're emotionally attached to. うち the adjective specifiying which of the noun is closest to you.
So for this sentence, you closest or immediate family.
Or those words are homynyms. I don't know the kanji for うち the adjective.
The kanji for the うち that means home/house is 家.
The kanji for the うち that's being used here as a pronoun is 内 (literally means "inside", which links to Mortal Canem's comment above about the concept of うち referring more to a concept of us as in "me and my inner circle" rather than just a general "us")
From what I understamd though, the Japanese people generally use the hiragana うち rather than the kanji when they're using it as a pronoun. They use the kanji 内 more often when they want to actually convey the literal idea of "inside".
I was under the assumption that かぞく was understood as your family. if you have うちのかぞく would that not literally be "House family"?
Never in my three years of living here in Hokkaido have I heard someone describe "our family" in this way.
Seems a little odd to be teaching Kansai slang specifically