What does it mean to let gluten relax?
It means exactly what it says: the gluten relaxes. You can observe it for yourself.
When you leave the dough alone after vigorous kneading, the gluten relaxes and goes from a tensed up state to a mellow state, similar to how human muscles relax depending on the owner's emotional state. The dough goes from hard and modeling-clay-like to pliable and elastic.
The difference for the final bread is a bit indirect. While trying to bake with a completely tense gluten will probably lead to a bad rise, you actually would never do that, because you leave the dough to rise after kneading anyway, and it relaxes during that stage. The real advantage of giving the dough time to relax during kneading is that it lets you properly knead the dough. If your dough has tensed up too much, further kneading motions are not making the structure better, they are making it worse - breaking the dough, for example, and certainly not contributing to the pliable, resilient elastic mesh of gluten you want. If you try to force tensed up dough by kneading, you will end up with overkneaded dough, and if you stop at the point that it's tensed up and let it rise, it will be underkneaded. So instead, you give it a rest, the dough relaxes, and from that point on, you can continue kneading.