If a guy isn't super serious about you, there's no way he'll introduce you to his family. The last thing he needs is to have his mom asking after you or his siblings talking about how cool you are. When you get invited to meet the family, he's indicating that he thinks you're special. When he wants you to continuously hang out with his family it means he is very much in love.
So, if you're in love too, go to as many of those family dinners as possible. Even if his family isn't the best, it will show him that you love him right back. Becoming part of the family is probably one of the most significant steps to becoming a serious couple that really goes the distance. We can already hear the wedding bells!