If I had eight hours to chop down a tree, I'd spend six sharpening my axe. - Abraham LincolnIn other words, you need to spend time on things that make the other things you do more efficient. And you need to do this even while other emergencies are constantly creeping up. The deadline is approaching, but you still have to do the things that let you do the things...faster.
Now in literal terms, I don't know how to sharpen a saw. I know how to sharpen a knife, or other smooth-bladed implement. A saw is probably similar, but I don't know that for sure.
The metaphor is essentially that when you have a large task, represented by the tree, you have to do things that seem to be unrelated (the saw sharpening) in order to cut down the tree. The saw works well as a metaphor because a dull saw WILL cut down a tree, but a sharp one will cut down the tree faster and with less effort on the lumberjack's part.
As an example, for homecare. Let's say you have a washing machine. It's kind of old, so it breaks down. You now face a choice. You can spend three hours fixing it. Or, alternatively, you can wash your clothes in the bathtub, taking 30 minutes per time. (Whereas before you needed maybe 2.) The productivity experts are urging you to fix the washing machine as soon as possible, because there will always be emergencies, and the extra 28 minutes add up pretty quickly.
This doesn't, however, mean that you can avoid tasks you need to do. Not washing the laundry at all in my previous metaphor could probably last a while if you have a big wardrobe, but while you're not washing, you have less and less to wear and more and more sweaty-old clothing, and if you ignore it enough, it does become an emergency. (If for no other reason than you either have nothing clean to wear, or because the massive pile of laundry now blocks access to what you need.)
I now wonder: In your trade or field, how can you "sharpen" your "saw"?