Why does the beef in Goulash need to be floured and brown before cooking?
Flouring the beef does two things:


It facilitates the development of a crust on the outside of the meat that is brown and flavorful;
It contributes to the development of the sauce, as the starch will dissolve off of the meat into the sauce, where it will swell at about 180 F, helping to thicken the sauce.


In this particular recipe, the flour is the only source of thickening, so it is being used for both purposes, but mostly for the second, where it is essentially creating a roux as your brown the beef, because you are also cooking the flour in the olive oil.  In this recipe, omitting it would give you a much thinner sauce.