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 cannot turn down these assignments just because they don't want the case or don't like the client. Courts need lawyers to step up and help people who need representation, so judges expect lawyers to accept these appointments in most situations.
There are only three good reasons a lawyer can refuse. First, if taking the case would force the lawyer to break ethics rules or other laws, they can say no. Second, if the case would cost the lawyer too much money to handle, that counts as a valid reason to decline. Third, if the lawyer finds the client or the case so offensive that it would hurt their ability to do a good job representing that person, they can turn it down.
This rule exists because everyone deserves legal representation, even people who are unpopular or facing serious charges. Lawyers took an oath to serve the legal system, and part of that job means sometimes representing clients they might not choose on their own. The rule keeps lawyers from cherry-picking only easy or likeable cases while leaving harder cases for someone else.