unfortunately, this sentence stands in isolation, so one really has to deduce everything from that which follows:
最近になってやっと、家族の大切さがわかってきた。
I feel petty confused about the part before the comma xD I have a strong feeling some stuff was omitted there ^^
I only dare to make an attempt at translation this far: "..., I came to understand the value of family."
最近になって means "becoming very recent", "arriving at the recent past", or simply "very recently".
やっと means "finally", "at last", "eventually".
家族の大切さがわかってきた means, as you say, "I came to understand the importance of family".
Put them together and you get "At last, very recently, I came to understand the importance of family". More idiomatically, "It was only very recently that I came to understand the importance of [the] family".
