Anything that happens significantly happens in a big way. A significantly happier person has gotten much happier. Moving into a significantly bigger house means you can finally get that indoor pony.
Significant things are important. When you see the word significantly, you know something is a big deal. A significantly higher tax rate would upset many people, because taxes went up a lot. A significantly lower-than-usual snowfall would upset skiers, because no one likes skiing on dry ground. A significantly timed speech happens at a time that is meaningful for some reason, like on a major holiday. This word often means the same as importantly.