What advice would you give to the spouse/children of a newly enlisted soldier, on how to deal with understanding what it's like to be away?

The family needs to be told the responsibilities of the Solder, one of which requires leaving for extended periods of time. Explain the importance of school/training or the overall goal of the deployment. Being gone is part of the job that the Soldier agreed to do. For deployments or long missions, encourage the family to attend briefings; gives them a chance to hear someone else explain why the Soldier is leaving and gives them a since of community seeing others in the same situation.
The spouse at home needs to have his/her own identity (outside of military spouse). He/She should have support persons/friends that could be relied upon to vent without judgement. Get involved with FRG or some sort of group that actively gets together and does activities. Have a plan in place to pay for bills (auto pay, list of due dates, online passwords, etc.) - sometimes the simplest thing are forgot when you are stressed. Also have a plan to keep in touch - emails, letters, skype.. and possibly a count down or calendar for children.

Be honest. Tell them about your work and how it is important. Tell them you miss them when you are gone. If you do deployments, send letters, talk through the internet, do Skype or Messenger calls. Tell them you have friends, tell about the dog or cat that hangs around, about the food, the weather. Keep communication flowing. Don't leave your family wondering how you are doing or if you think about them. TELL THEM.

Best bet is set them up for success, you cannot "explain" it away.
You can only prep them for things that may come up.
What is dads role, tasks, job today....who will fill those roles while you're gone?
think of the what if's and if you would be the one to address them, who will now...Do not do any of that in a vacuum ..but think though first then discuss in family meetings.
Empower the kids as age apropret to self solve/support. Empower the spouse the same.

Not just a list on the fridge, but an actually trial run/meeting of support agencies. If a water pipe breaks... not just that number in a note book... but walk though the process...to include how to pay for that water heater and install christmas eve.. you cant afford it, thats fact (if it is) so how? figure it out now, not on her at 22:00 at home, with no hot water and kids needing baths.