If you joined a lobby since patch 8.5, then congratulations—you were invited to, and successfully attended, a party! Parties are persistent lobbies with integrated voice chat (as of patch 8.6). An open party means anyone from any teammate's friend list can join automatically, without asking for an invite. Whether this is a good thing is up to you, because you can also set up a closed party by clicking the slider in the top right of your lobby.
Once you're in a party, you'll be able to use voice chat. It's on (and voice-activated) by default, so if you're already on Discord with your team, you'll hear some echoes. You can open voice chat settings with the mic icon in the bottom right of the client, next to missions and chat. This is where you'll go to turn voice chat by default on or off (off, probably), and set up push-to-talk buttons. You can also access these settings from the regular settings menu.
Some answers to some questions:
Why can we only talk to our premades? I want to ask this random jungler a question.
There is a camp who wants voice chat to call shots, MIAs, and ganks. There is also a camp who doesn't want to deal with voice chat. It's hard to fully satisfy both camps, so for now we're only allowing voice chat among people who party up.
Voice chat with random people has some powerful drawbacks--namely, that anyone can say whatever they want to you, and often do. Pair voice chat with a hyper-competitive game like League, and you've got a recipe for conflict and stress that some people just want to avoid. We could have made voice chat opt-in, but that creates a lot of pressure for players to opt in for the sake of competitive advantage. We're going to keep voice chat parties-only for now.
Can I report someone who's been awful in voice chat?
We don't have any behavior systems in place for voice right now. Since it's premade-only, you'll have to hash out your issues with whatever friends or friends-of-friends you're partying with.
I have a lot of friends that I might not always want to play with, can I opt out of open parties?
Yep. Just have the party leader click the slider in the top-right of your lobby to remove the green checkmark.
I'm having an issue with voice chat =(
We've got a troubleshooting article for that. If you've dealt with voice in games before, the troubleshooting steps will look pretty familiar.
I'm using Windows XP, why isn't voice chat working for me?
Because you're using Windows XP. It's older than a lot of players, so go ahead and update your OS.