"Who, where, when and how will the decision be made if the victim, who lost will have to pay these costs or not."
In a personal injury case, each side pays its own attorneys' fees. HOWEVER, the prevailing party is ALWAYS entitled to recover its costs. I recently finished what was effectively a personal injury case (not really, but close enough) that went on for a year and a half. The injured plaintiff is going to end up on the hook for over $23000 in costs (mostly costs associated with taking depositions and obtaining copies of medical records). The statement in the retainer agreement is simply an advisor to you about this.
"My mother is permanently injured after a slip and fall accident and signed that kind of agreement. We are worried that we will lose our entire savings, income etc. and have debts in extreme high amounts."
Even if your mother were to lose and get hit with a hefty judgment for the other side's costs, "we" wouldn't lose anything. Only she would be liable. You would not be liable.
"How can we protect us from the liability of potential opposing party's attorney's fees and costs?"
Again, there is no "we" or "us" here. If your mother is the party to the lawsuit, she is the only one with any risk. And, as noted above, liability for fees isn't possible.