How can people in this situation afford lawyers?

Those lawyers only take cases they are sure they can win. They go after a lump sum settlement that is equal to the amount the disabled person would have collected from the beginning. They usually keep 1/3 of that for their fee as soon as the government pays it.

People who live on disability struggle because the benefits are low.

The lawyers take a percentage of the lump sum settlement -- a nice large percentage. It often takes a long time to get a social secuirty disability claim approved. Pretty much automatically gets turned down the first tiem. Unless you have a very drastic disability claim, it can take years without a lawyer.

When finally approved, social security back pays to the initial claim. So that initial lump sum payment can be several years worth of monthly benefits. The lawyer takes a huge chunk of that.

Neither - the lawyers work on contingency which means if they win, they take about 33% of whatever is won, if they lose no one gets anything. Basically, these attorneys can do this because they either do class action suits (many people, lots of money), they are banking on settling out of court (little work) or they have everything down to a formula (again, boilerplate with little time invested).

EDIT:

Jamie would be wrong - these are civil cases and the government does not appoint lawyers to act for plaintiffs in civil cases - they appoint lawyers to DEFEND low income people (people are entitled to counsel in their defense).

The claimant does not pay the lawyer up front. The lawyers take case on contingency. Their fee is paid only if you win your case and it is set by the government at around 12%. Thats why they advertise and why they take on so many cases at one time. And why you dont hear from them much after they take your case.

If people have low income, or can't afford, then the court appoints them a lawyer for free. These lawyers are paid for by the government. Of course, you cannot guarantee that these lawyers will win you the case. The best lawyers of course charge expensively.

Do not listen to Steve D's answer; sometimes that happens, but most times it's the above scenario.