There's nothing wrong with side of things.
The implication behind the expression—in this case—is that there are two types of problems: those that are technical and those that aren't. People who address a problem can be thought of as being in one of two groups, and on "separate sides" of any given situation.
In the example sentence, the person being asked to contact technical support has more technical knowledge and will be better able to understand a technical conversation. They know more about the technical nature of the problem than does the speaker. (An unstated part of the sentence is a final "than I do.")
This particular expression takes the plural things because there are many components of the issue, some technical and others non-technical. (And, grammatically, if it were singular, it would be side of the thing.)