This summary is not the rule. The rule text controls.
If you want to join someone else's lawsuit, you have to follow a specific process. You need to file a written request called a motion to intervene, and you must send a copy of it to everyone already involved in the case, following the delivery rules laid out in Rule 5.
Your motion needs to explain why you have the right to join the case. You also have to attach a separate document that lays out your actual claim or defense, basically showing the court what side you're on and what you're asking for. This gives the other parties a clear picture of your position, not just your reason for wanting in.
This same process applies even if a specific law gives you the right to intervene. In other words, having a legal right to join a case doesn't skip the paperwork. You still need to file the motion, explain your grounds, and attach your claim or defense just like anyone else asking to join.