Our solar system formed as a nebula collapsed into a solar nebula. The collapsing gasses and dust began to spin and flatten out. Pockets of denser gas and dust collected due to gravity and became planetesimals. Planetesimals are building blocks to planets. Think Lego blocks to a Lego castle. Their mass means gravity. As gravity increased they attracted more material. The gravity becomes so great at some point that gasses like nitrogen and oxygen form an atmosphere. More mass means lighter gasses like hydrogen and helium can be held. Mass isn't the only factor at work though. Temperature plays a role as well.
The four Jovian gas giants are far from the sun. This means that they are cold. Cold gasses have less kinetic energy which means they have a hard time breaking away from an atmosphere. That's why Earth doesn't have much free hydrogen or helium. Our mass might be enough to hold some of it, but we are warm so those gasses have enough energy to break away from out atmosphere. Saturn has both a large mass and cold temps. It can hold on to light gasses. As it attracted those gasses it became more massive which increased its gravity. More gravity means more gasses to attract. . . and so the cycle went until Saturn attracted all that it could, which wound up making it way more massive than earth. Not more dense though. Interesting Saturn fact - it's less dense than water which makes it equivalent to a planet sized beach ball.
Did you know that inside of Saturn is a rocky, core that is about the size of Earth? So without all of the gas surrounding Saturn, we would be about the same size. So why is there tremendous amounts of gas surrounding Saturn and not us? Why aren't we the size of Saturn? Well...
There are two different types of planets in our Solar System: Terrestrial and Jovian. Terrestrial are found inside of the Frost Line (an invisible barrier outside of the Asteroid Belt where Hydrogen isn't vaporized due to the heat of the Sun) and Jovian outside of the Frost Line.
When our Solar System was created from a planetary nebula (smaller cloud of gas in space), the planets were formed by rotating gas getting squeezed together by gravity through a process called accretion. All planets were accreting material, but soon the Terrestrial planets ran out of material to grab on to. They could only hold on to rock and metal. However, the Jovian planets were free to accrete the gaseous Hydrogen gas and become the massive planets we see today.
So to answer your question, Saturn is bigger than Earth because of it's position from the Sun, the chemical properties of Hydrogen and the Frost Line.
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