"Larry is a big family man and he was only too happy about that."
To say that someone is "a big family man" usually means that he places great importance on his family and enjoys being involved in family matters.
For example, from Yahoo.com/questions: "I'm a big family man and I'm wanting to become an ER doctor. Will I have time to be a good father?".
In your example, without more information we can only speculate about why he was happy, but it might be in a context like this. "Last Christmas, Larry's brothers and their wives and kids all had dinner at his place. Larry is a big family man and he was only too happy about that".
Here, "only too" is an emphasis, Larry was very happy about it, whatever it was.