This summary is not the rule. The rule text controls.
When a court asks a lawyer to take on a case, the lawyer usually has to say yes. Lawyers can't just turn down an appointment because they don't want the case or don't like the client. Courts need lawyers to step up and represent people who need help, including people who can't afford to hire their own lawyer.
There are only a few good reasons a lawyer can refuse an appointment. First, if taking the case would force the lawyer to break ethics rules or the law, they can say no. Second, if the case would cost the lawyer too much money to handle, that's also a valid reason to decline. Third, if the lawyer feels so strongly against the client or the case that it would get in the way of doing a good job, they can ask to be excused.
Outside of these situations, a lawyer is expected to take the case even if it's inconvenient, unpopular, or not something they would normally choose to work on. This rule makes sure people who need legal help can get a lawyer, and it stops lawyers from dodging hard or unpopular cases without a real reason.