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Mostly through taking cuttings from the mother plant and attaching them to another plant in a process called grafting, which allows them to keep growing.

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There are three basic ways to do this.

For long-lived vines, bushes, and trees, it's done by cuttings. You find a rare mutant plant that produces good fruit with no seeds, then take cuttings from it. You root those in soil or graft them on to roots, and you essentially make a bunch of clones of the original. This is how it works for bananas and grapes.

For annual plants, it's done by hybrids. For example, seedless watermelons are made by hybridizing two strains that produce infertile watermelons with few seeds. It's a bit like breeding mules.

Finally, in some circumstances you can get seedless fruit by preventing fruit from being pollenated. This is how pineapples are kept seedless, for example, and it's why Hawaii was a major pineapple producer. There were no native pollinators that would pollenate the fruit, resulting in seed free pineapples.

That was really interesting, thanks.

Basically, you make a copy of the tree (or bush or vine, whatever)

What you do is you plant any old seed for that fruit until you get a seedling. Then you take a living branch from the tree that makes the seedless fruit and you replace the top of the seedling with it. This is called grafting. Basically, the seedling will repair itself with that new branch and continue growing from that new branch until it becomes a fruit bearing tree, making new seedless fruit. The beginning stages of the graft are the most dangerous because the seedling basically has an open wound and a very weak connection, but once that heals over its just like caring for a normal tree.

This is exactly what we have to do for apples because when apple trees reproduce, their offspring is very different from its parent, and very unpredictable. It makes it hard to get good apples from seeds, so we use grafts to essentially cline existing trees that we know make good apples. This also means that most fruit you eat is genetically identical (ie all honeycrisp apples are the same, all strawberries are the same)

This does lead certain varieties of fruit to be prone to disease. For example, the dominant banana in supermarkets in the 50s was the Gros Michel, and it tasted like the banana flavoring we have today (because the flavor was based on that variety of banana) but most of the trees died out due to a fungus that specifically attacked that tree. Bananas we have today are a different variety called the cavendish banana. They could easily fall victim to a similar fate, because like other fruits, they are all genetically identical to each other. We do have other varieties of banana produced in smaller quantities that are prepared to take over should the cavendish die out.

Wait, pineapples have seeds?! Damn, you just messed my head real good

well, most of them don't, but they would if they were fertilized. It would look like this

Interestingly, pineapples almost never reproduce by seeds. Even in the wild, most of their reproduction is vegetative. Why? Their huge, spiny fruits are adapted for being eaten by the extinct megafauna of South America. Ground sloths, Gomphotheres, etc. When those animals went extinct, pineapples and their relatives lost their usual means of dispersion and had to limp by on occasional lucky seed spread and vegetative growth...until humans started growing them.

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The trick they do with seedless watermelons is really neat.

Basically, they have strain A of watermelons that produces seeds. They take it, and make strain B by doubling up every chromosome. The chromosomal composition of B is the same as A, so the resulting plant behaves the same and still produces seeds. You can then make offspring plants with one A parent and one B parent, taking half the chromosomes from each - so it gets, for instance 15/30 from A and 30/60 from B. The resulting plant is still chromosomally equivalent to A and B, but it has an odd number of chromosomes - 45. So when it goes to create sex cells (which grow into seeds), it can't, because those require that the chromosomes be evenly split. So it grows fruit that are identical to A/B, but that don't produce seeds.

Sinsemilla has entered the chat.

With cannabis there are female plants and Male plants. In the old days you planted seeds and waited to see if it had Male tops then pulled up all the males. Then the female flowers won't get pollinated, no seeds, sin semilla in Spanish.

Nowadays it's all grown from clones, cuttings that are rooted. All female.