From what I understand, the fundamental idea behind the Spaceborne Computer is to forego most of the expensive, time-consuming, one-off, bespoke hardware modifications commonly used for radiation hardening and other "space resilience", and instead recover resilience and reliability using software.
The stereotypical "space computer" uses manufacturing technologies that are decades old because current structures are so tiny that they are very susceptible to harsh environmental influences like radiation. It also uses bespoke hardware built for this specific mission. Then it spends many years being designed, verified, and tested. So, by the time it becomes operational, it is already 20 years out-of-date. And because it was so expensive to build, it will then be run for 15 years to recover its cost.
This is heavily exaggerated of course, but if you look at the computers that are currently on the ISS, it is highly likely that at least one of those things will be true for each one of them.
The Spaceborne Computer, on the other hand, is mostly a standard off-the-shelf computer. The only really bespoke part of its hardware is how it interfaces with the ISS, i.e. the shape of the case and how it is mounted, electrical connectors, integration into the environmental systems (waste heat), and data connections.
The hardware, however, are essentially standard Hewlett-Packard Enterprise servers and edge computing devices (HPE Edgeline Converged Edge and ProLiant DL360), pretty much identical to the ones used in some supercomputers. In addition to the capabilities of its predecessor, the Spaceborn Computer-2 also has multiple GPGPUs to accelerate image processing, video processing, and machine learning / AI workloads.
One of the experiments is that the SBC-2 will also apply machine learning to itself to be able to learn and adapt to anomalies.
Question: What "advanced computer capable of faster processing and data compression" is this?
The trick is that it is not actually that advanced. But, because it is a current generation, commercial off-the-shelf computer, it is much more advanced than computers that are purpose-built for space operations.
The setup costs for producing a current generation state-of-the-art high-end CPU are insanely high. Somewhat exaggerated, you could say that producing 1 CPU or 100 million CPUs basically costs the same, because the setup costs dwarf the per-unit material cost. Therefore, you can only afford to build high-performance CPUs if you sell huge quantities of them, and that is simply not the case for purpose-built "space CPUs".
For a similar reason, most supercomputers in the top 500 list now use off-the-shelf hardware instead of highly specialized hardware: gamers simply outspend governments due to their sheer numbers, so manufacturers can actually afford to invest more research into gaming hardware than supercomputing hardware. And thankfully it turns out that there is a huge overlap in the computing requirements of games and scientific computing – much of scientific computing is about simulations, and what is a game other than a simulation?
So, the simple answer is that the "advanced" computer is actually a "normal" computer, but until now, the computers that were in space were not "normal".
Another benefit is that because it is (supposed to be) so much cheaper, you can afford to replace it every 2-3 years, much like you would a COTS high-performance computer on Earth.
What aspects of its faster processing and it's data compression require space-testing?
The fact that its hardware is not actually designed for space. Rather, Hewlett-Packard is hoping to compensate for that by layering a software layer of reliability and resiliency on top of the potentially unreliable hardware.
what is it about this computer that can improve "time-to-insight" by this factor?
This is essentially just an application of the idea of "edge computing" to what Hewlett-Packard calls the "ultimate edge" in one of their press releases. The idea behind edge computing is that in many modern systems, the bottleneck is actually not the computing power, but rather moving the data to where the computing power is, i.e. network bandwidth. What edge computing does, is to put computing power where the data is: on the factory floor, on an oil rig, etc. (Similar to how more and more intelligence is pushed into phones and away from servers, or "Internet of Things", where sensors bring their own computing power with them and make decisions on their own instead of just being dumb raw data streaming devices.)
This is even more extreme in space: communication bandwidth as well as latency are severely limiting. If you could number crunch your terabyte of measurements in space and then simply send the result back to Earth instead of having to send your terabyte of raw data, you could get a huge improvement in time from experiment to result. It might be faster to process the data on Earth in a giant data center, but the processing time saved might still be lower than the transmission time.
This might not be as pronounced on the ISS, but in addition to this computer actually being used on the ISS, this is also an experiment in itself with the goal to investigate putting supercomputers on the Moon, on Mars, and on deep-space probes.
How can it be so much better at processing and compressing data than existing computers on the ISS?
Because those are old and this one isn't. That is really the simple answer behind all the marketing.