What makes oil stick to the pan so bad that it is so difficult to wash out?

As I understand it, when you combine high heat and vegetable oil you get a fairly stable polymer (much like a plastic or resin). The polymer bonds with the surface (which is porous), and results in robust surface. This is what we use to season cast iron cookware, but it's less desirable on stainless and other lighter coloured hardware.

You can prevent the coating from forming by never allowing he oil to maintain a high temperature for long periods by itself. Keeping food in contact with the pan will both help keep the temperature down, and reduce the chance of the oil bonding with the steel.

Cleaning the oil off once it's formed is usually pretty simple: add water (or soda water) and heat for 10-15 minutes. Often the oil will wipe out (or need mild scrubbing with an appropriate scrubbing pad). You should not have to use steel wool (or a caustic cleaner), both of which can damage the smooth surface of many pans.

If you get oil too hot, it can form a really sticky layer of the heavier oil residues.

To remove it, use a little fresh oil, warm the pan, and spread the fresh oil all over the surface. Let it rest for a while, then spread the oil on the surface again. Tip the oil out, rubbing away as much residual oil as possible with paper.

Now you can rub neat dishwashing liquid onto the surface and all the residual oil will emulsify, and can be washed away with water.

What type of oil did you use in cooking? There are Drying oils, Semi-drying oils, and Non-drying oils.

Drying oils are polyunsaturated materials such as linseed (flax), or Tung oil. People usually use oils like these to finish furniture, but some, like walnut oil, do get used in cooking.

Semi-drying oils, like corn, sunflower, safflower or soybean oil, have high enough unsaturation to oxidize into a gooey mess on your cookware, especially if you get them too hot.

Non-drying oils, like olive, canola or peanut oil, are relatively saturated and don't easily oxidize to form the sort of hard varnishes you have to clean off off with a scrubby pad or angle grinder.

If you cook with non-stick pans, you should also be careful to read the instructions on the pan. Mine says to never heat above medium. My last pan was ruined after heating above medium with a black substance much like yours.

Here's the other matter. After you've cooked with oil in a pan, mine is the one with the dreaded circular groves, as in Circulon® and any residual oil remaining will turn into the thick sticky substance after at least a day. My mistake was not washing it shortly after using the pan(skillet). If you use a quality oven cleaner, that will take care of the problem, but be sure the manufacturer thinks it's OK for its cookware (or if you know it's scientifically alright to do so without damage to you or cookware). Be careful of the fumes.

Oil, butter, wax and greasy substances like that are all hydrophobic, meaning they do not mix with water. The following site can explain this. https://study.com/academy/lesson/what-is-hydrophobic-definition-interactions-quiz.html

Use a baking soda paste to clean the baking tray.