I doubt whether you have worded your question correctly as according to your question the answer should come out as 576 and not 288. Anyway an approach that could lead to 288 is as follows:
Let's number girls and boys as : G1, G2, G3, G4 and B1, B2, B3 respectively.
Since a girl has to sit next to a girl only therefore all the girls would sit together G1,G2,G3,G4 or G2,G3,G1,G4 etc. Now these girls can be arranged in 4! = 24 ways.
Now we have 3 boys who can be made to sit together in 3! = 6 ways.
Finally we have Girls (G) and Boys (B) who have to be seated together and they can be seated in 2! = 2 ways (consider 4 girls as one group = G and 3 boys as one group = B)
From above three steps we get the total number of ways as : 4! * 3! * 2! = 288