I've said this before in this space, but the Earth is weird. Consider this:
These plots show the distribution of surface elevations on a bunch of large and small bodies in our solar system. Something should jump out pretty quickly: All have a single peak for elevation except for Earth and Mars. That's weird! Why would any planet have two different peaks for the distribution of elevation?
For Earth, the explanation is relatively straight forward: we have two types of crust. Oceanic crust that underlies the oceans is thin, dense and low in silica (silicon oxide) while continental crust is thick, buoyant and silica rich. This creates a planet where much of the land surface lies beneath sea level (in this case, 0 kilometers elevation), creating ocean basins. The rest of the planet sticks up to 8 kilometers above sea level, creating the continents.
Our planet's surface is made of tectonic plates, slabs of crust and upper mantle (the next layer down). Depending on who you talk to, there are potentially dozens of major, minor and microplates on Earth. They all interact in different ways to create or destroy crust and rearrange themselves.
What we consider continents aren't directly equivalent to the tectonic plates of the same name. North America (the plate) contains both the land mass we think of as North America (the continent) as well as the oceanic crust that runs from the eastern seaboard to the Mid-Atlantic Ridge. This means that many plates are a combination of oceanic and continental crust.
Alright, so why does this make Earth weird? No other planet* has such defined binary crust types that creates the twin peaks of topography. Something different is happening on Earth and it has been happening in different ways for potentially over 4 billion years (*we'll come back to Mars soon).
Young Earth Started the Continents
The early Earth was a very different place. Go back 4 billion years and the Earth's surface was mostly basalt -- the stuff of oceanic crust. After the formation of the planet (and the subsequent formation of the Moon by a large impact), the planet likely had a magma ocean that solidified into dark basalt.
The Earth was much hotter back then as well. That heat was coming from the leftovers of the planet's formation and all the impacts that happening during the early history of the solar system. There was also more heat being generated by radioactive elements that have since decayed.
In fact, it was hot enough inside the Earth that when basalt cooled and began to sink, it started to melt. When you only melt a little bit of rock, you don't get magma that is the same composition of the rock you are melting -- a process called partial melting -- because only certain minerals melt. This means that the basalt melting on early Earth was creating magma that was much more enriched in silica, just like our modern continents.
This may have been the birth of continents as we know them on Earth. These areas of silica-rich magma cooled into rocks like tonalite and granodiorite (close cousins to granite) and suddenly the Earth's surface wasn't just basalt. Over periods of hundreds of millions years, this churn created vast bodies of silica-rich, buoyant new crust.
This process didn't last forever. As Earth cooled, the dominant process to form new crust changed to plate tectonics. There is evidence that 60-70% of the Earth's continental crust was formed by around 3 billion years ago. It continues to grow today, but at rates that are more than 3 times slower. Most of that growth is happening where plates are colliding to form subduction zones where oceanic plates are shoved back into the mantle like the Cascades or Andes.
Mars is the only planet with two peaks in elevation, but the reason for Mars' split is controversial. It might be tectonics, but there are theories that massive impacts on the southern or northern hemisphere could have created the elevation differences between the hemispheres. It could even just be a product of our accounting of surface elevations on Mars, a world without any oceans today.
Of course, the big "wait a second" question here is why our sister planet, Venus, never really developed distinct continents. There is some suggestion that an area of high elevation (Ishtar Terra) could be something akin to early Earth "continental" material, but without exploring the Venusian surface, it is just speculation. In any case, Venus isn't Earth's twin by any means.
The Earth may have just had the right balance between heat, water and formation history (like the lunar-forming collision) to start the ball rolling towards continents. It is hard to imagine our planet without the highs of continents and lows of oceans. World lacking such features exist out there and any life that arose on them might be very different.