Not sure I follow.

The question here is asking for "moles", not molality

"How many moles of AgCl are contained in 244 mL of 0.135 m AgCl solution?"

I know that the question is asking for mols of AgCl, but it says that the solution is 0.135 m. Correct me if I'm wrong, but I was taught that the lowercase m represents molality.

Therefore, the solution is 0.135 moles AgCl/1 kg solvent (because molality is moles of solute/kg of solvent)

In the solution, they multiplied 0.135 m * 297.68 g, where 297.68 g is the kilograms of the SOLUTION.

The problem is that 0.135 m is equivalent to 0.135 moles AgCl/1 kg SOLVENT.

So, multiplying 0.135 by 0.29768 kg is multiplying  0.135 moles AgCl/1 kg SOLVENT *  0.29768 kg SOLUTION.

So, the units do not quite cancel out. I was thinking about how I can some how find the kilograms of solvent from the kilograms of solution, but I am not sure how to do it.

I know what you mean, it is ambiguous, but in terms of this question the m stands for molarity, that is, mol/L. Otherwise you're right.

But technically it should be capital M