It's subtle. You can't get it from a dictionary. More context would be helpful but it is clear enough to a native English speaker.
The two phrases "getting over" and "getting through" are often used in the connection of an emotionally difficult life event--the death of a close relative, a breakup with a boyfriend or girlfriend, losing a job.
"Since we broke up, I feel as if my life is over... I don't know what to do."
"I know it seems hard to believe now, but you WILL get over it. Trust me about this."
The writer wants us to pay attention to a subtle difference in the two phrases.
To "get over things" suggests just passively waiting it out. The terrible feelings will just go away with time.
To "get through things" suggests DOING something active, to "work through" the situation in some way.
"Kubler-Ross talks about 'grief work.' I worry about your spending so much time at home just sitting alone. You need to do something active to get through this period in your life."