and I've read that Hubble uses a digital camera to take pictures like a cell phone.
I can't find a place in nasa.gov that says that the Hubble works like a cell phone, but of course the principle is the same.
An optical system takes parallel light coming from various directions and focuses each parallel bundle to a small point.
We used to say this is how a camera works back when cameras were big and had obvious lenses on one end and big rectangles of photographic film on the other, but most people just use the tiny camera inside their phone.
Sources: left, right
The Hubble telescope is also a camera but with a much bigger lens. It is built a lot more like these; the first two are cameras, the third is a telescope without an imager. From this answer to Can an astronomical telescope view objects on Earth? which basically explains that a telescope is just a big camera :-)
Sources: left, right
Is the digital camera built to see billions of light years away, or does Hubble have some technology on it that allows it to zoom into the deep space?
There are four things that allow the Hubble space "camera" to photograph objects so much far away, and therefore so much smaller and dimmer than we can with our cell phone cameras.
- A really really big aperture, 2.4 meters! It's much bigger than say a 3 to 5 mm aperture of a cell phone camera.
- A really long focal length. At f/24 and 2.4 meters, it has a focal length of 57,600 mm, which is ten thousand times longer than the f=5.6 mm focal length lens in my cell phone.
- A really good CCD which has been optimized for this application
- It's in space, above the Earth's atmosphere, so there is no wiggling or astronomical seeing problems that we have on Earth.
These days people can image large astronomical objects at night with cell phones because the tiny cameras are improving and especially because the computers inside the cell phones are improving allowing for hefty, substantial amounts of image processing to make good looking but not necessarily scientifically accurate images. See for example: Milky Way with a cell phone, how is this possible? in Photography SE.