The recipe linked to the question is not very detailed. It does not even say to beat the egg whites (which you did and which needs to be done). Here is a snippet of another recipe:
You just need a sugar thermometer so you can cook the sugar until it reaches a temperature of 113°C/235°F. Once the sugar syrup reaches that temperature, immediately take it off the heat – or you’ll risk bringing the sugar up to ‘hard-ball’ stage – and carefully drizzle it into the foamy egg whites, mixing continuously. You do remember the foamy egg whites, right? I bet you do. Drizzle the hot (!) syrup slowly into the egg whites, making sure not to pour it directly onto the whisk attachment of your mixer, or the syrup could splatter against the sides of the bowl or into your face.
I think the key is "drizzle into, mixing continuously". And here is a picture of what it should look like at that stage (from the same source):
[OP} But since I don't understand the chemistry of egg whites I am not sure how to figure it out.
There is some chemistry and some physics here. You want to add a lot of sugar and little water. You need the sugar in solution so you can add it quickly, and if the syrup cools down, it is no longer a liquid. On the other hand, you don't want the egg white to heat up too much, so that's why you drizzle in the syrup while mixing vigorously. That way, the syrup cools down while being diluted with the water in the beaten egg whites, effectively removing water from the bubbles surrounded by the protein from the egg white, making the foam more stable.
If everything else was correct (water content of syrup, using bowls that are not greasy, correct ratio of egg whites to syrup), I think the key is to drizzle the syrup into the egg white where it gets mixed right away. You describe adding it to the side of the bowl, but the recipe does not say that. If I were to experiment with the recipe, this step would be my focus.