Oven spring is caused by the air pockets in the dough expanding from the heat. (Dough rises from gasses released from the yeast.)
After the shaping and final rise, often times there is a light, dry "skin" over the dough. By slashing a dough before it goes into the oven, you break this skin, and the bread is able to expand. If the loaf is a "fancy loaf" and you can't slash it without ruining the appearance (like a braided loaf), try to keep the loaf from drying out with a light mist of cooking spray after shaping and before the final rise.
Perhaps the bread is cooking too quickly when it hits the heat of the oven, essentially cooking a crust before the air pockets get heated enough to expand? Baking in a moist environment should help with that. Place an empty, sturdy pan (I use a cast iron skillet) on the bottom rack of the oven (or directly on the floor of the oven) before preheating. When you place the bread in the oven, pour about 1 cup of very hot water into the empty pan. This will create a bunch of steam, and help prevent the bread from crusting before it gets its "spring."
If you have a pizza/baking stone, use it. Having a hot surface to set your pans on helps with the rise. Think about it... you open the oven door, and out goes a lot of the heat... even though the walls of the oven are retaining the heat, it will take a little while before that heat reaches the bread. Setting the pan on the stone will give you that instant heat on the bottom, causing dough to rise from the bottom up, rather than just getting a small rise from the top area.
If you don't have a stone, invert a heavy baking sheet, cast iron griddle, or something similar, and heat that up in the oven the same as you would for a baking stone. Invert the sheet pan so it's easier to slide your bread pans on and off the hot pan without having to deal with a small edge.
If you haven't got equipment to do the other options listed above, you can try the "cold oven method." Just put the loaf into a cold oven, and set the temperature. Don't preheat. The gradual heat from the bottom of the oven as it preheats will give you some of that "oven spring."
I use a baking stone and steam, but I have had great success with using an inverted aluminum sheet pan with steam before I got the stone... and before I learned that trick, I used the simple cold oven method (no steam as the oven is cold!).
I mix dough in my kitchenAide all the time, and this happens all the time, so through the process I check the consistency and dryness and I've developed a feel for the "just right formula", but being ADHD I don't go in with precise measurements each time, but that would be a good idea once perfection is found. So anywho, I take a strong spatula from time to time and cut it of the hook, to give it a nice restart with hook on outside of dough ball. but your dough needs to be dry enough that it is cleaning itself off the bowl and it will the hook too. It's designed to find that perfect balance so that it starts wet and sticky and then as the gluten starts to web it becomes self cleaning, and the kneading is drying it a little or maybe the material is absorbing the liquid, either way it goes from dry to wet. Another thing I do if the hook is wrapped in dough instead of working against the sides, I briefly turn up the speed and that will throw the dough to the sides and sometimes clean it.
Best Answer
Besides the initial gluten relaxing that mrwienerdog wrote about, another important effect is the absorption of liquid.
It takes time for the flour to fully hydrate with the available liquid. Before it does the dough will be more sticky and there will be less gluten available for kneading. Giving the recently incorporated dough a short rest allows the flour to pull in all the liquid it can and become much easier to work with.
A similar effect can be seen with pastry doughs. Even though often the goal is to minimize gluten development, a short rest gives the flour time to hydrate and can turn pastry dough from difficult to work and crumbly to soft and smooth.