Speed and reliability. Take open source. You could manually find, download, untar, configure, make, install and track dependencies. A good package manager lets you do that in one command e.g.:
$ brew install package_name
OSX includes a package manager for proprietary packages blessed by Apple. It's fast and reliable but its curators forbid most open source and proprietary packages. It's called App Store.
Ignoring the problem of software availability, the App Store also lacks a fast command line interface. I can kick off a homebrew package install process and return to whatever I was doing faster than the App Store.app launches.
If you're coming from a linux environment, you're probably used to having Apt or Yum available. OSX has no built in package manager so you've either got go out and find compiled binaries somewhere or install from source, both of which are tedious and searching the web for it is time consuming vs. the `brew install wget` commands that you can probably guess and it'll take care of installation in the background while you do something else.
No, Yum isn't available or compatible, it's coupled to Red Hat/Fedora/CentOS linux distributions like Apt is coupled to Debian/Ubuntu/Mint distributions.
Homebrew/MacPorts/Fink are nearly the same thing, they just don't come installed by default on OSX.