Russell maintains that universals subsist in the Problems of Philosophy. Before I explain the meaning of this expression itself, I have to define what each of these two ideas mean.
First, Russell characterizes universals as non-particular. Universals, according to him, exist as qualities and relationship between things. Three reasons given in a logical order prove their being. The first reason is that a thing could not possess a quality unless there was such a thing as this quality. Then there would be no way to know that this thing would possess this quality unless we were already familiar with it. The final reason is that a quality would be meaningless if there were not anything for which it arose. An example given by Russell about it is the notion of justice. If someone asked you what justice is, you would think about the principle or ideal of just dealing or right action which is considered as its very own essence. You will find yourself "looking for something which all just acts have in common; this thing which all just acts have in common is, naturally, what we mean to get at when we speak of justice" (p. 16). These qualities and relationships between things are what Russell labels universals. They are anything with characteristics shared by many particulars, and those characteristics serve to distinguish justice from acts or even what is a piece of wax.
Now, we have to consider what subsistence means. The strict definition is that subsistence is the condition of remaining in existence. In more explicit words, in general, it is considered as more suitable to say that things exist when they are in time, things like physical objects or feelings. However, universals subsist because they are opposed to existence. They have beings, they are eternal. They do not depend on time or humans to remain. Even if humanity disappears and the earth implodes, a triangle will always be a geometrical figure with 3 sides.