How are volcanoes and earthquakes linked to plate tectonics?

First, you have to understand what convection is.

Did you know that the hotter something, the more it expands? This happen because heat, at an atomic level, is nothing but vibration. If a liquid is hot, it means that there is more vibration between it's atoms. And, more vibration causes to have more space between the atoms, which causes the density to decrease.

So, when a liquid has it's top cold, and it's bottom hot, the top will be more attracted by the gravity than the bottom, because it's more dense. This will cause the top to want a lot to get down, and the bottom, because of the pressure of the top, will want to go up. This will cause a spiral movement; the bigger the difference, the faster and the more violent this spiral movement will be (and that's what causes hurricanes)

So, what this has to do with Earthquakes and stuff?

The Earth, when it was formed, was really, REALLY hot. It has being losing it's heat ever since, but, since there is not air in the space, it can only lose it's eat though radiation, which is very  slow. And also, the bigger something is, the slower it cools down, because there is more vibration, more movement, more energy to be lost. The earth was formed 4,5 BILLIONS of years ago. That's A LOT of time. But the lost of heat by radiation is so slow, and the earth has so much heat to be lost, that it still didn't cool down that much.

You can think of Earth like a hot potato that got out of the oven just a little ago; the outer part is already cold, but just take a bite, and you will see how HOT it's insides are. You can't take a bit out of earth, of course, but it's insides are a lot hotter that the outside, because it's the outside that loses eat. The insides have to wait for the outsides to cool down, and then lose eat to the outside.

So... we got a planet that's really hot in the inside, so hot that it's mantle is made out of liquid rocks!

So, now, it's the time to connect the dots: the upper part of the mantle loses heat faster than the inner one. This causes a spiral movement, and, since the tectonic plaques are floating in this liquid, they move as well!

When two plaques move one against each other, they crash against themselves, and this causes an earthquake. If they move away, this creates an open space; and since the liquid of the mantle has, like, THE WHOLE CRUST OF THE PLANET CRUSHING IT, the mantle is craving to get out, so when there is an opening, it squirts out like crazy, creating an eruption. After a while, the magma solidifies, creating a volcano.