Why should one pre-heat an oven?

It is not a problem to put whole potatoes for roasting in the oven while it heats up. In fact, it's a good idea.

However, if you are following instructions or cooking times written for a preheated oven, you will need to add about 10 minutes to the cooking time (depending on how fast your oven heats up).

The rule of thumb I use for this is to add 1/2 of the time my oven takes to heat up to the recommended cooking time. Since my oven takes about 20 minutes to get to 400F, that's 10 minutes.

However, regarding bell peppers it depends on why you are putting them in the oven. If you're talking stuffed bell peppers, then it's fine to start with a cold oven. However, if you are trying to char the peppers to remove the skins, you need to start with an oven that's as hot as possible, otherwise you will cook the peppers to mush before the skins are charred (and, ideally, use the broiler).

FuzzyChef gave a good answer. To add to this a bit, the reason it's good to cook potatoes in a cold oven or starting from a cold pot of water is so they cook more evenly. Otherwise the outside is more likely to overcook while the inside is still raw. It's more important in the case of boiling potatoes than in the oven which is a gentler cooking method.

I'll also add that it takes most ovens quite a while to heat up and stabilize. I don't think this really matters so much for potatoes or especially bell peppers, but you definitely want to preheat for any kind of baking unless the recipe specifically says otherwise.

If you are trying to char bell peppers, I'd suggest a pair of metal tongs over a gas range or a blowtorch if you have either of these. Or a grill. Or a baby dragon.

The other major reason to heat the oven in advance is that the oven will run the heating elements at full duty cycle until the oven comes to temperature. This effectively turns the bottom of your oven into an upside-down broiler. While the overall oven temperature is still rising, radiant heat from the full-power element on the bottom will be blasting the bottom of whatever food is sitting on the rack and you risk burning the bottoms of whatever is in there.

The food will be blocking the element from heating the top of the oven and will be soaking up that heat instead - all on the bottom of the dish, which can get much hotter than it should, and hotter than the oven's set temperature during warm-up.

Once the oven is warm the element cycles on and off in short waves, meaning your food tends to be more surrounded by a uniform ambient temperature rather than being blasted with highly directional heat from below. Most recipes expect a uniform heat, and the initial ~10 minutes of uninterrupted broil during heat-up can destroy whatever it is that you're cooking.

For the exact same reason why it is highly recommended to put meat in boiling water if you favor "lesso" (boiled meat) while you should put it in cold water if you want a tastier broth.

Thermal shock tends to form a surface crust on things you are cooking, preventing fluids to spill out.

This is much more evident with bell peppers than potatoes, of course.

If you are cooking them together in a cold oven you may end up with a "potato in pepper juice" mess that will (eventually) dry up, but will cook at 100°C for a long while, which is not what You want.

Of course this depends very much on how much time your oven take to heat, but even a few minutes can be fatal.

Preheating is very important in some cases, and in other cases it will not matter much at all. What matters is how crucial time and temp are to the thing you are making. One thing that it will always get you is consistency, which can be a good thing for inexperienced cooks because it eliminates the starting temperature variable from the equation. That factor is less important as you learn to cook by feel instead of religiously following a recipe.

That being said there's a pretty easy rule of thumb for when it is most crucial to follow the instructions -- is it something you would buy in a bakery? Does it seem 'delicate'? Then stick with all the preheating and such because it will probably be very important. Is it something like a simple baked pasta dish, potatoes, or a frozen pizza? Not going to matter as much so long as you know when to pull it.

An exception that is worth noting: do you want your homemade pizza to turn out like the fancy wood-fired joint down the street? Then put your oven as hot as it can go and leave it there for a literal hour: it absolutely will make a huge difference. But this follows from the time and temp rule, a homemade pizza greatly benefits from being cooked as hot as your oven can go for a short amount of time.

Basically all good answers, especially about the "Thermal Shock" creating a crust, to which I would add, due to caramelization and Maillard reaction, which is a primary flavor/flavinoid source.

The MAIN reason however, is that to achieve high vitamin, high protein, flavorful food that is still low in fat and salt, it is SUPREMELY ADVANTAGEOUS to move starches, proteins, and vegetables into FAT only when the FAT is very hot to avoid the food absorbing excess FAT.

The heat reservoir created by hot oil or lard allows the food to cook, often reaching several tens of millimeters into the interior and browning the outside in a very short time exposure to the emulsified, liquid fat, which can be absorbed into the food material. The longer you cook at a low starting temperature, with an insufficient heat source, the more fat the food absorbs.

Basically the same for an oven, you will render more fat into the pan and have the right amount of butter, lard, or other oil browned and absorbed into the food combined with the natural striping or stipling of fat in meat.