In Oil Paints

What part does linseed oil play in oil paints?
Linseed oil, when mixed as an ingredient in artists' oil paints, is known as a BINDER. It 'binds' the other ingredients of the paint together, both while in paste or liquid form, and while drying after use. Why is it needed? Paint consists of some kind of coloring agent, usually finely-ground pigments of some mineral. This fine dust needs something to make it wet so it can be spread around, and needs something to keep it uniform, 'floating', so it doesn't all clump up and settle to the bottom.

For a more modern example, think of Ice Cream; the different ingredients have different weights, and without a binder they would slowly settle into different layers, the lighter fluid on top, then the heavier creams and sediment on the bottom. So a binder (usually commercial-grade guar gum) is used to bind all those ingredients together and keep them floating, rather than settling into layers.

In ancient oil paints, egg yolk was used (this process and product was called TEMPERA, or TEMPURA); you've noticed that, when you eat cooked eggs, they very quickly dry and form a hard shell on your plate... or when you throw them at cars and houses at Halloween, they stick like crazy and are responsible for much grumbling and swearing by whoever must scrub them off. They are very tough, but have some poor qualities; as times evolved, a binder was needed that had better properties for oil painting... dried slower, was easier to use, didn't rot as fast as egg yolk before use, dried harder, didn't change color over time, etc. Linseed oil was found to possess the perfect qualities, and has now been used for hundreds of years for this purpose.

The properties that linseed oil adds to oil paints are desired and powerful:
It keeps a perfect consistency in the paint pigments, not too runny and not dry or clumpy; it comes out of the tube or bottle in just the way it went in.
It keeps the paint moist for a long time after use; the artist can revisit the painting and make changes again and again, for days or weeks, while it slowly dries. Different processes and additives can give linseed oil different properties, so it can take many forms and purposes. Linseed oil mixtures are used to make oil paint harder, softer, give it different thicknesses, make it dry faster or prolong the drying times.

Linseed oil has a wonderful behavior of seeping, or leeching, into most surfaces or materials it comes into contact with, so rather than just 'sitting on the top and drying', it helps bind paint to canvas or wood, helps bind new paint into old paint, and helps blend one color with a different color very subtly; a perfect 'nexus'.
Its 'oily' quality, even when dried, keeps colors looking vivid, kind of like when you rub a little water over a pebble; the dry pebble doesn't have much color, but when there's water on it the color becomes very visible, vivid. Linseed oil makes this effect permanent. And this gentle clear 'sheen' is very similar to the outer coating of many things... human skin, plant skins, water surfaces, so the use of linseed oil in oil paints has made oil painting the most desired art form to closely reproduce life.