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There are pretty straightforward physical “rules” for how the size of an explosion translates into how much damage is done, how big the fireball gets, how radioactive it gets, how tall the cloud goes, and so on. (All of these, except the radioactivity, apply to non-nuclear explosions as well.) So what they did was have lots of ways of measuring the effects of the blast: high-speed cameras that let you see how big the fireball was; airplanes that could measure the height of the mushroom cloud; blast-pressure sensors that could keep track of how powerful the blast wave was; and so on. Then they would work backwards from the data and their models to estimate the actual yield.
There are a few tricky things. These multiple methods didn’t always match up, both because nature is more complicated than simple equations (the humidity in the air, for example, can affect the blast distances, as can the temperature of the ground or water), because their measurements could have imprecisions to them, and because different nuclear weapon designs could output the energy in slightly different proportions (e.g., more as heat than blast, or more as radioactivity than blast). So sometimes they didn’t get it quite right, or came up with conflicting estimates — the radiochemical yield would be different from the thermal yield which could be different from the blast yield which could be different from the fireball size yield. They’d look at all of these and throw out the ones that looked sketchy and just decide on a number.
There has been a lot of work by the national labs over the last decade or so to digitize and re-analyze some of the high-speed photography taken of the tests with modern computers and physics models, and they’ve found that some of the tests were different yields than they expected (I think the max variation is 10%, which isn’t a whole lot, but could still matter for some types of calculations regarding the accuracy of weapons needed to destroy a given target).
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