Nobody’s really backed up here and answered your question fully.
In general, getting from your device (PC, phone, tablet, etc…) to a site on the internet is really a matter of routing messages from one network to another, to another, etc…
In your house, you usually have one network, your Local Area Network (LAN) that connects all of the devices in your home. It connects them to each other- they can all talk to each other on the LAN. That’s usually how network printers work- they don’t go through the internet- your PC or phone or whatever sends network messages directly to the printer through your LAN. In most homes these days, the LAN is wireless, and is called “Wi-Fi” (it’s usually some flavor of 802.11 wireless ethernet).
Usually one of those devices is something called a router- it basically acts as a connection point between networks- you’ll see it on a windows devices as your “Default Gateway” in most cases. What it does is forward any traffic that one of your devices wants to send outside of your LAN- i.e. anything outside of your family’s devices on the LAN out to another network- in this case, your internet provider’s network. Their routers do the same thing- forwarding that message until it ends up where it is supposed to be.
There’s also something called a ‘modem’, which is short for modulator/demodulator, and its’ the device that translates the signals so that they can be sent via a phone line (DSL) or an analog cable TV line (cable internet). The modem’s just a physical layer gizmo- it doesn’t do anything but change one type of electrical signal format into another one.
Most people have a single network appliance that does multiple things- it’s a router, it’s a Wi-Fi hub, it’s a DSL modem or cable modem, it’s a rudimentary firewall (something that limits access from the outside world), etc… It’s the gizmo that your internet provider gave you that hooks up to the wall.
Based on the article, it looks like the Wi-Fi access points they’re talking about are essentially little cellular routers that have their own Wi-Fi network and forward onto a carrier’s cellular network (like 4G LTE or 5G or whatever). Same basic idea as what your internet provider’s gizmo does, except that it sends/receives external traffic over a cellular network like a phone instead of through a DSL line or cable line, so for the purposes of providing fast, cheap internet access to low income students, it’s the best bet. This is because it provides adequate speed, and requires no additional hardware or infrastructure beyond what the cellular phone carriers have already provisioned to most places.
Of course, if said low-income students live somewhere without cell coverage, they’re not going to be able to use it.