To try it out, open the Spotify app on your iPhone (it's seemingly an iOS exclusive for now) and tap the magnifying glass (Search) icon in the bottom row. Then, you should see a circular white microphone icon in the bottom row.
Tap that, and then tap OK to grant Spotify access to your device's microphone. Now you're able to start vocally asking things of Spotify. Also, once this is enabled, you can hold down on the search icon to give voice commands.
In my testing, I found Spotify to be hit or miss at properly recognizing my intent. It properly processed commands such as "play Rap Caviar" (a popular Spotify playlist) and "play songs by Katy Perry," but it failed at other requests.
For example, asking it to "play songs by Battles" (an American experimental rock band that is in Spotify) opened up the Epic Rap Battles of History playlist. Saying "play songs by The Red Hot Chili Peppers" produced an error where it couldn't find them, as it turned out asking the same request, minus the "The" was the correct way. Siri caught both of those requests correctly, for what it's worth.
But as you've probably noticed, there's an even bigger annoyance here: you can't open Spotify's voice controls without opening the Spotify app. And, even then, you need to tap to activate voice mode. With Apple Music and Siri, I can just ask my phone to play songs by Busta Rhymes, and it just works.
According to Casey Newton at The Verge, the feature started appearing in the iOS Spotify app recently, so Android users may be waiting longer. In Newton's article, he also referred to the feature as "an experiment," so its early bugs may be attributable to it being in a nascent, beta state, though the Spotify app makes no such qualifying statements about it being unfinished.
If Spotify's voice control hasn't rolled out to you yet, you have two options. Try force-quitting the app and re-opening it — this can sometimes trigger an app to refresh and access new features — or simply wait.
As an Apple Music user who holds onto his Spotify app for checking out exclusive playlists (and days when I need to report on it), this lackluster version of voice control is slightly reassuring to me. Yes, I've chosen a platform that many of my friends aren't on, but at least it's the one that's got the best integration with my iPhone.