An Idea of how the energy field circuit board works

The dreams that the three characters (later five) have are actually transmitted instructions on how to build the circuit. The circuit board itself is not a generator. Instead, it's a data transceiver analogous to a cellular phone. Like a cell phone, it's actually a low-power radio transmitter and requires a cell tower (with its larger antennas and powerful radios and high-bandwidth data/voice lines) to function.

When the board, connected to the Apple IIc, is powered on, the transceiver joins the network. Like a cell phone it gets identified and queried.

At this point, the computer begins "programming itself" What's actually happening is that whoever or whatever is managing the network has queried the new device and is pushing the software needed to generate and control the energy field. Once the software is running, the device is ready for use. The Apple IIc is simply the interface. Like Android is the operating system that controls the baseband processor (which has its own OS and communicates with the cell towers).

We might have actually seen the actual machinery that generates the field on the ship the three encounter. The large chair with various bubbles is the dream transmitter. All the supporting machinery we see in the guts of the ship could actually be what generates the field. During a scene, Wolfgang mentions that the energy field is extraordinary but needs more work and mentions "talk about bumming rides" which could mean that the ship actually generates the field.

Consider that when the ThunderRoad's field fails, I originally thought that it was because the main battery failed. The battery was a salvaged 12V car battery that apparently wasn't kept charged. But Wolfgang and Darren were the types to make sure that it was charged before a journey. The actual cause of the failure might actually had been the aliens' father shutting the field generator down. An improper shutdown can cause other equipment connected to it to fail such as a surge or simply a loss of data. In either case, that might've caused the IIc, its power supply, or the board to fail.

In which case those three are lucky they walked away from that crash.
When the dreams resumed, it meant that the aliens either escaped their father or someone else reactivated the machinery.

The field generator itself could be anything. It could be mobile or stationary (relatively stationary). It could be large or small. It could be on the kid's ship or something they tapped/hacked into.

One last note. The addition to the circuit board that provided them with oxygen could have been some kind of exit point for a worm hole. The added circuit could be analogous to a beacon. This is where the wormhole exits.