There's already a lot of great commentary on this forums that lends itself well to an in depth understanding of the subject.
If you're looking for a simpler explanation, I'll try my hand:
Consider that the two formats of photography (Film and Digital) rely on different properties. Film, like digital, is sensitive to light, but through a chemical processes rather than a digital one. That is to say, there are chemical reagents that coat film and are reactive to light.
No surprises here. In black and white film, these light sensitive reagents are made mostly of grains of silver. So, if an engineer wanted to make a film more sensitive to light, they would need to add larger grains of silver to the film coating. The graininess one sees in a film photo is quite literally these grains of silver showing up, since they've reacted to the light! So, it should be no surprise that films with larger grains of silver (high ISO films) appear more grainy relative to films with finer silver grains.
The digital process works in quite a different way. On a digital sensor, what we refer to as "grain" appears as a byproduct of something called a "signal to noise ratio". An analogous effect can be found in your car's radio. Consider, when you're downtown near the radio station, the radio signal is strong (all signal, no static noise) and your tunes sound clear. But, head out to a rural area and you'll hear a lot more static (Less signal, therefore more static noise). The same is true for a digital sensor! In low light, you'll find that there is less signal (light) for your camera to detect. By amplifying this signal (high ISO values) you are lifting put all the background noise as well. That is to say, we have a lower ratio of signal to noise information. You can imagine this noise is like a form of visual static, this is where you'll see grainy images on digital cameras. So, if not visible light, what is your sensor picking up on? Invisible light! These other parts of our whole electromagnetic spectrum which we understand as light. For example, the infrared spectrum (what we feel as heat is in fact a form of light) and the Ultraviolet spectrum what gives us sunburns despite being invisible to our eyes is also light. Now, this can get a lot more complex from this point on, since both of these processes are controlled by clever ways by camera and sensor manufacturers, but for the sake a keeping it simple, just understand that the Chemical Process and the Digital Process produce their grain in different ways.
Now, this explanation is much simpler that the complete process which takes place in our cameras every day. Some parts of this explanation may be misleading as well, since there are chemical and electromagnetic processes at work in both film and digital. But, I hope that you will find that this simplified explanation is helpful in understanding the basics of your question. There are certainly more in depth explanations that can be found online and will be there for you to seek out when you're ready. All the best!