"Significant cutbacks" mean that more than just a few people were fired. If 1% of the employees of a company were laid off, you'd call it "minor" cutbacks. If 40% of employees got laid off, you'd call it "major" cutbacks. But something between maybe 5%-15% would probably count as "significant" cutbacks.
"Significant" is not only used with cutbacks, of course. It's used to talk about all kinds of numbers and percentages:
A significant number of the people we asked had never even heard of Twitter.
There's been a significant drop in the number of complaints since the new system was put in place.