As the name suggests, Voice over LTE is what happens when your carrier allows you to place a phone call over your LTE connection instead of the more common voice networks. Verizon Wireless, for example, traditionally used 1XRTT for all of your voice calls, relying on LTE for data. This used to be why Verizon phones couldn't simultaneously use voice and data. AT&T and T-Mobile, which relied on a combination of LTE for data and HSPA+ for calls, would drop down to a 3G signal when talking to someone on the other line. With VoLTE, neither of these scenarios is necessary anymore.
What both network types now have in common thanks to VoLTE is the ability to use more bandwidth to make phone calls with higher quality audio traveling both ways. When you are on a call with someone else who is using VoLTE, you immediately notice the difference in call quality on both ends. You'll also notice calls connect faster if you are calling someone near you, and while that's not a feature worth bragging about, it's a cool thing to test for yourself.