If you're on the verge of crying, you're not crying, but you're almost crying. You might go ahead and cry, or you might recover yourself and stop the tears from flowing, like this girl does. To be on the verge of tears is to be really close to tears.
If you're on the verge of shouting or yelling, you're not doing it yet, but you're about to. If you're on the verge of understanding something, you don't quite understand it yet, but you're getting really close! And if you're on the verge of winning, you haven't won yet, but you're about to.
We often talk about people being on the verge of a meltdown, or on the verge of a nervous breakdown. In fact, if you just say you're on the verge, it means you're on the verge of melting down, freaking out, or otherwise panicking.
You can also say that one thing is verging on some quality, meaning it almost has that quality, but not quite. For example, turquoise is a shade of blue verging on green: it's getting pretty close to green.