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Do you mean the big white space all around the image? Or do you mean those white horizontal strips? Oh no, I see. The white strips are on purpose, so you mean the white around the blue.
Without knowing specifically which options you used for saving and exporting, my best guess is that probably rogue invisible rectangle, that you might not even be aware of.
I found it by Select All > Ungroup, ungroup, and ungroup, until there are no more groups left. The status bar will tell you when there are no more groups left.) Then without deselecting, and possibly you might need to zoom out, you can see that the bounding box (dashed line rectangle) goes around some empty space.
Now deselect everything, and start tapping the Tab key, slowly. This selects one object after another. When that rectangle which has no fill or stroke is selected, you can just delete it. Oh, actually it's selected on the first tab for me.
I'm not sure why saving the file included that invisible rectangle, because it's completely outside the page border. Some image viewers will show what's outside the page border, and some don't. So I would guess that the saved images are correct. And yes, that's one of the few complaints I've had about Windows Explorer, the thumbnail shows the whole image, ignoring the page size. I don't know, maybe there's an option somewhere that I've never found.
But anyway, for exporting the PNG, you need to pay attention to the top 4 buttons in the export PNG dialog. For this, you should use either the Page option, or select the image (excluding invisible rectangle), and use the Selection button. Either of those should do what you want.
If that info doesn't solve the problem, or if I've misunderstood, please add more details about the white space.
Edit - Actually if you were saving as PDF, or other format, there might be other options that you need. If so, let us know which format, and we can tell you which option you need.
