What does "through the bag" mean?

Think of a golf bag as a series of 10-12 different tools for solving a problem. There are many people who are really good with one of those tools, so much so that they use it in place of several of the other tools. This is a strategy, and it can work for some people.

But for most people, their best strategy is to get as good as they can with all the tools. This means building core strength for many of the irons, fine motor skills for putters, problem-solving (and occasional brute strength) for wedges, and solid aim for woods.

This is what it means to play through-the-bag, to utilize all the clubs for their designated uses and work through their individual peculiarities, rather than becoming super good at your nine-iron in hopes you can skip having to learn the complexities of wedges and woods.