Krdytkyu

The classic advice given to backpackers trying to limit the weight they have to carry is “Pay attention to the ounces, and the pounds take care of themselves.”

Brooks Landon M.A. Ph.D.. Building Great Sentences: How to Write the Kinds of Sentences You Love to Read (Great Courses) (2013).

How's the quoted sentence true in reference to backpacking specifically?

The pounds cause most of the mass and physical burden.

It does make sense to watch the weight of your gear and go for lighter gear when possible. A few ounces here and there will add up to pounds.

On the other hand, it is also possible to be penny wise and pound foolish and in my experience at least, this is far more common, you will see overweight hikers and climbers buying titanium gear for example.

I once had another backpacker with his gut hanging over his belt inform me that his water shoes were 4 ounces lighter than mine.

Realistically, this is just a saying and not an ironclad rule.

There are several expressions akin to

Pay attention to the ounces, and the pounds take care of themselves.

The most famous saying is

Take care of the pennies and the pounds will take care of themselves.

ascribed to Lord Chesterfield in 1747 by Phrase Finder. (There were 240 pence (pennies) to the pound sterling.) The statement means "to be thrifty and not to squander money", and was memorably updated by Sen. Everett Dirksen, commenting on how federal spending tended to get out of control:

a billion dollars here, a billion dollars there, and pretty soon
you are talking about real money.

Ounces or pennies or a mere billion dollars, the principle is the same: look at the cost or weight of every item you buy or take with you on a backpacking trip and try to pare down the weight or cost and your wallet will be fatter and your pack lighter.

Some backpackers (and racing sailors) cut the handle of their toothbrush in half to save weight, but you don't have to be that extreme. Some mountaineers make one of their water bottles do double duty as their pee bottle when they are tentbound, but IMO, that is going much too far. Carry extra clothes so you can change into dry clothes, but don't worry about dirty clothes. Repackage food to eliminate glass and cardboard. Take a tiny travel sized tube of toothpaste (preferably already partially used) -- or no toothpaste at all, just floss -- your teeth won't drop out in a couple of weeks.

Everything you take should be essential, according to your standards, and everything should be scrutinized: (1) do I really need X? (2) do I really need this much of X? (3) Can I repackage X to eliminate weight?