Is that because mother's milk doesn't have enough salt?
In short, yes. It's a problem mostly in places where the mother's milk doesn't have enough of pretty much anything, but salt's the one that kills first.
Consider a place where an average salary is $40 a month. Unfortunately, there are millions of people (infants and mothers included) who live where half of that would be considered a wealthy income. Surely you've seen the desolate scenes on TV where they ask for some number of cents per day to buy little Mary a pair of shoes to walk over the rocky debris to school... We're talking about those places, and worse.
These are places where having clean water isn't as great a concern as having any water. Most of the local population is undernourished, including the mothers. Without proper nutrition, they produce too little milk, and what they do produce is too poor in nutrients to support the infant.
From a biological perspective, salt is fascinating*. In the body, it serves to provide many of the ions needed to control molecules, and it holds water in various places. That's why eating salty food makes you feel dehydrated - your salty blood pulls water from the other tissue. Similarly, when that salt makes its way to your urine, more water is pulled with it, making you urinate more (spawning many myths (and some facts) about salty drinks cleansing the body).
In an infant with a salt deficiency, the lack of salt prevents the intestines from working properly, as the cellular channels lack the energy to open. That prevents nutrients (including salt) from being absorbed into the blood. The blood's low salt level stops the absorption of water, leaving the feces liquid, which will quickly be released, carrying the vital salt with it. Where an adult would be able to hold their stool in longer or try to eat more food to compensate for the lower absorption rate, an infant can't do that of its own will, and the mother can't just produce more milk on demand, especially if she's also undernourished.
The cure is a solution - one of "clean" water with salt and sugar (as fuzzyfuzzyfungus noted above), that can easily be absorbed, raising the blood's salt level, allowing more nutrients and water to be absorbed.
If I had known the cure were that easy, I would have told more people. One problem is that people just don't know that is the cure (even if they are worried about diarrhoea as an issue)
Unfortunately, it's also not as easy as telling people on the Internet about the condition. People with access to the Internet aren't likely to be affected by it. It is pretty common knowledge among related volunteer organizations, but there is a severe lack of knowledge in the local communities where the problem is deadly. There are many medical volunteer groups, and they do great work... but the problem is bigger than their limited resources can cover.
* My biochemistry knowledge is remembered from five years ago. The facts presented may or may not be entirely true.