How do I use a remote raid pass?

In order to use a Remote Raid Pass, the raid must show up in your "Nearby" tracker. The range is not infinite, remote raid passes will not let you see raids you couldn't see otherwise.

Under the icon for the raid in the Nearby tracker, a small "View" button will be visible that will take you to the gym, similar to when you click on a defender pokemon to visit the gym it is in. Once at the gym, the "BATTLE" button will be purple and say

BATTLE
^{using a Remote Raid Pass}

Click the button and it will put you in the lobby.

EDIT: A feature to invite in-game friends has been added. Once in a lobby there is a small button with an icon that looks like "plus person." If you click on this, it will let you send an invite to anyone on your friends list. They will receive the invitation in their nearby raids tracker. If they click on that invitation, they will be brought to the gym and allowed to use a remote raid pass to join the lobby. Be aware that the invite only lasts until the lobby runs out of time, so if you send invites with only a short time left your friends may not see it and react before the fight starts, locking them out of the lobby.

The remote pass lets you participate to a Raid remotely (range is not infinite either, but it's still rather large like maybe 500 meters or so), but a Raid must still be on-going. They are still popping up as they always did, which means that there isn't going to be any after 9 or 10 PM at night for instance.

Now it is also opened up for the possibility to invite friends to a raid. That is if one of your friends is at the location of a raid and has entered the lobby, he/she can invite you to that raid.

To accept the invite, you will have to use a remote raid pass.

For an invitation like this, there are no limit on distance.

See official info from Niantic