For most smaller nonprofits, accepting donations of securities can require lots of paperwork and incur expensive transaction fees. With a DAF, you can donate the security to the DAF, have the DAF sell it without tax consequences, and then have the DAF donate cash.
You can donate appreciated assets (such as stock) to a charity directly, and you get to deduct the current market price on your taxes without having to pay capital gains on the asset.
Donor advised funds simply make this process simpler. You can donate your appreciated asset to a donor advised fund (which acts as a 'charity') and immediately realize the tax write-off, and then later decide what charity to give the money to.
This article seems to be making a big issue out of the fact that there are a lot of donor advised funds in Silicon Valley that are waiting to give the money out, or giving it out over time instead of making one big donation immediately. I fail to see the issue - this is just people moving money around to optimize for taxes - the money will still be given to a charity.