Why do base kilometers help with time trials?
It’s all about maximizing the positive effects of training while minimizing recovery time and injury risk
You want to target different systems in the body as effectively as possible. A naive approach to training would be to (try to) train at race intensity and duration each and every day. However you’ll quickly reach a point where you can’t recover fast enough, suffer from injuries or reach a point where you don’t improve anymore. Different systems in the body also respond to different intensities.
Long rides are great to strengthen the heart muscle, improve oxygen uptake of the muscles and improve how fast you can burn fat. Training these systems mostly depends on training duration and less on intensity. Long and easy can even be done after hard training days and shorter easy rides can improve recovery because of increased blood flow through the muscles. Short, high intensity training is best to improve other factors like VO₂Max or pure muscle strength.
Apparently until a few decades ago athletes used to perform the same workouts, day in day out, the whole year long. Periodization was kind of a break-through. Now you generally start the season with a lot of long, easy endurance workouts and replace them with more and more relatively short, high quality, high intensity workouts the closer you get to the final race day.
Edit: I’ll try to answer the follow up questions:

I remember an episode on That Triathlon Show where an expert cited research that cool-down (i.e. continuing at very low intensity after a hard workout) actually doesn’t have any positive effect for recovery or training effectiveness. She mentioned that the only positive effect is that it prevents people from fainting after a race because it prevents the blood from suddenly pooling in the legs (I imagine simply lying down would achieve this as well). Not sure why people still seem to do it, maybe because it’s also a good way to calm down.
At least for running, Daniel Craig’s ”Running Formula” recommends to do running if you want to get better at running. I think he mentions that off-season activities like cross country skiing can have a positive effect but he doesn’t recommend it if all you want to do is get better at running. I think the same applies to bicycling though since it’s less dependent on form/technique it probably benefits more than running from exercise which targets similar muscles, endurance etc.
I don’t know about that one. I think your question boils down to if doing a certain type of workout for a few days or weeks in a row (e.g. several weeks of easy rides during base training phase) is more effective than simply throwing it in every now and then. Some training plans do have supercompensation blocks where you do high intensity workouts for a few days in a row (more or less intentionally going above the maximum recoverable volume). Not sure you could fit that in a monthly training cycle.