Keep reminding yourself that it is your choice to go and study. Look at all of the positive experiences you have had since you started. Reassure yourself that it is OK to feel so good about home, and so sad when you leave it. It would be a very sorry day for you if all of your family experiences were so negative that the very thought of home created no emotion and just despair. For some this is a genuine reality.
Remind yourself too that you can't live at home forever but your head will always be with your 'home' and your family. One day you may be providing that 'home' too for your own family. This is just one of many exciting stages for you in a lifetime of finding out about the world and the people in it.
Its OK to feel like you do. It is lovely that you love your family so much to feel like that. Just roll with the emotions and stay positive.