Here's what gain of function means in a biological context. When you have a mutation occur to a protein coding gene, one of two overall things can happen - nothing or something. Genetic code has some built in redundancies that mean that you can swap out some DNA base pairs and end up with the same protein in the end. If something happens, there's broadly two types of thing that could happen to the resulting protein. You could have a loss of function - where the gene basically breaks, or a gain of funciton - where the gene either works better or gains the ability to do something new. (Of course "nothing much happens" is also an option here if the new protein is similar to the old one).
One important context this comes up in is cancer. You have genes in your genome that are actively stopping cancer from forming and spreading (tumor suppressor genes) as well as genes that would, if allowed to work unrestrained ( think a gene that signals for cell growth) would cause cancer (proto-oncogenes). A gain of function mutation in a proto-oncogene turns it into a cancer-driving oncogene, and a loss of function gene in tumor suppressors opens the door for oncogenes to do their work unrestrained.
In the context of this virus, the authors identified a mutation in the spike protein (one of the important proteins in the virus involved in a lot of its biological processes including infeting a cell) that is not present in it's evolutionary cousins. Basically, this particular coronavirus spreading right now, even though it is very closely related to previous coronaviruses, seems to ahve this one bit in its genetic code that is starkly different, suggesting that this strain/subspecies/species picked up this mutation recently. Now, seeing that the new coronavirus has this genetic feature could indicate one of two things. Either, it's just an artifact of the founders effect where the small population of viruses that jumped from that bat or snake or pangolin to a human would have genetic differences just by chance, and because that small population got blown up to a huge population, the mutation is just along for a ride. Or, it could be a factor in allowing this species to succeed in the first place.
OK, now to the actual mutation. The authors argue the latter - that this mutation might be important in the virus finding the success that it did. They basically found that this mutation leads to that part of the spike protein being more easily able to be cleaved (cut) by the protein Furin. This cleaving of the protein would better prime the virus to be able to infect cells and allow it to infect humans more specifically, and they note that Furin is highly expressed in the lungs as well. They note that a lot of experimental work has to be done to validate this, so it looks like they identified this computationally/with sequence analysis, but it could prove an important lead in terms of figuring out why is this coronavirus more infective than others in humans and provide a new therapeutic target for tackling this strain of the virus.
So to summarize, there are many species of coronaviruses out there, including many related to this particular strain. In this strain, however, there's a mutation that seems to be unique to it. THis mutaiton is a "gain of function" mutation, meaning that it allows the spike protein the mutation is in to work better or get new functionalities compared to SARS/MERS viruses. (If it was a loss of function mutation - then you could reason that you would get something less infective since the protein wouldn't be able to work). The hypothesized mechanism of gain of function in volves of protein highly expressed in the lung that could help explain why it's so much more contagious than previous coronaviruses and also provide a target for therapeutic intervention. More experimental work needs to be done to validate it.
I hope that was clear and noot too detailed or too high level - please let me know if any aspect was confusing and if that was a helpful explanation.