How can I get my elderly mother to throw out her old clothes?

You can't convince someone of this. They see their clothes as they remember them, rather than as they are. You can instead volunteer yourself to do her laundry (just do it, don't ask) and throw out a few of the old things, telling her later that they were old and just fell apart in the wash. Then, if she's up to it, take her out to find some replacement clothes. Or just present her with your peace offering of new clothes. She may be upset initially, but it usually isn't as painful to gain forgiveness, in these cases, as it is to get permission.