The short answer is she didn't have 200,000 warriors. That number is at best the Roman estimate of the total number of Iceni and other British fighting men in revolt, and at worst a total invention.
The longer answer does allow that the Roman force of one and a half legions plus auxiliaries was certainly outnumbered by the Britons, and whether it was three to one or the improbable twenty to one, it was still an overwhelming disparity.
Our primary sources for this are Tacitus, who was writing decades after the battle, and Cassius Dio, who was writing more than a century and a half later, so we have to accept that our picture of the conflict is imperfect.
We do know the uprising was widespread, and many Roman and Romano-British people died as the Iceni and like-minded Britons tried to purge Britain of the Romans. There was a climactic battle. We are not sure exactly where, although we are told the Roman governor Gaius Suetonius Paulinus chose his ground carefully. He positioned his army on a wide plain facing the mouth of a defile with a forest behind him. His thinking was to channel Boudica's forces into a frontal attack through the defile, limiting their frontage and ability to maneuver, while still being able to deploy his own (probably limited) cavalry on his wings, and with the forest behind eliminating the possibility of large masses of men coming at his rear in any kind of organized way. Tacitus likely invents a speech for Suetonius to give his men (I apologize for not having my copy of Tacitus open in front of me. I'm copying and pasting from Wikipedia, which says it's quoting Annals 14.36, but I'm not sure whose translation):
"Ignore the racket made by these savages. There are more women than men in their ranks. They are not soldiers — they are not even properly equipped. We have beaten them before and when they see our weapons and feel our spirit, they will crack. Stick together. Throw the javelins, then push forward: knock them down with your shields and finish them off with your swords. Forget about plunder. Just win and you will have everything."
Worth saying Tacitus's father-in-law was likely present at this battle, and while dialogue is usually invented in these sorts of histories, the battle as it unfolds may be based on a second-hand eye witness account. Anyway, Boudica led her army through the narrow defile in a frontal assault aimed directly at the middle of the Roman legionnaires' lines. The Romans did as they were trained to do: They threw their spears into the advancing Britons, wounding some and more importantly fouling the shields of many to the point where they were discarded in a hopeless tangle just as the charge was reaching the Roman ranks. The Romans advanced in formation into the tightly packed and disorganized Iceni and did to them as Romans had been doing to Celts since Gaius Julius Caesar started talking about all of Gaul being divided into three parts. Training, equipment, and discipline won out yet again, and Suetonius had enough cavalry to wrap Boudica's flanks and eventually harry her fleeing forces, who found their retreat hampered by the defile they had advanced through, and then further obstructed by the wagons in their camp.
As with so many battles in antiquity, the real casualty count begins when one side breaks and runs while the other pursues with organized purpose. That casualty count is then almost always multiplied by the victor to underline the scale of the victory. Tacitus says the Britons lost 80,000 while the Romans only lost 400. The Roman losses may well be accurate. The Britons' losses are almost certainly inflated, and maybe even by ten-fold. 80,000 may have been how many men Boudica actually brought to the battle. 80,000 may have been how many people —remember Suetonius saying there were more women than men present?— she brought, including camp followers and other dependents. 80,000 may be the total losses suffered by Boudica's side during the entire campaign, including those sold into slavery in the aftermath. 80,000 may also very well be a made up number to line up with the exaggerated 200,000 mentioned in the first place.
To summarize, you ask how Boudica lost with such an overwhelming advantage in numbers? The same way so many other Celtic war leaders lost after Rome implemented the Marian reforms in large part as a response to battles lost against Celts and Germans. Their way of waging war was not well-suited to beating a Roman field army of the 1st Century BCE or CE. The Romans had better tactics, strategy, training, equipment, logistics, etc. The uprising was doomed to fail as long as the Romans kept their heads and followed their own best practices based on hard-won experience.
Good afternoon, and thanks for the question! So, the short answer is that 1) 200,000, cited by Cassius Dio writing in the early 3rd century, is probably incorrect, 2) that the vastly superior training, skill and equipment of the Romans gave them a serious advantage when combined with Suetonius' careful selection of his field and Boudica's mistake of accepting battle in that field.
Dio's surviving Roman History is a composite piece to begin with, as the first 20 or so books are partially reconstructed from other sources that use him. Dio's numbers are always somewhat suspect as well, to the point that I teach my grad students to take his histories with a grain of salt. He writes very anachronistically here, of course, too, almost two centuries after the battle. It may have been a complete fabrication, or he may have described the total people involved. Tacitus, despite his love of inventing speeches (common to ancient historians, though maybe not here; the speech he recorded in Annals 14.36 is simple and to the point, and his father-in-law served here) is always a more credible source. He does give us numbers here, though, so we have to rely on Dio or modern, secondary estimates. Stephen Dando-Collins, in "Nero's Killing Machine," sticks to that number but Goldsworthy's "The Complete Roman Army" is more nuanced; he estimates 70,000 rebels, which is far more manageable.
The Roman Army gets a lot of attention, but it truly was an incredible fighting force at all levels, down to the individual level, as Ross Cowan points out in "For the Glory of Rome" and as Polybius explains. Here, it took advantage of a good commander and Boudica's tactical mistake of accepting battle in a narrow field to emphasize its training and equipment in a place where the numbers of the Britons were negated. They couldn't flank or maneuver at all, and had to go right at the Romans with inferior training and probably very little armor. Roman virtus made them extremely aggressive, but Suetonius seems to have kept his legionaries and Auxilia in check until it was time to press the advantage. The Romans were historically VERY prone of going berserk - see Josephus' Jewish War 6.257–58 - and tended to be uncontrollable when finally fully unleashed. The Britons packed their baggage and animals behind them and couldn't get out, so the berserk Romans slaughtered everyone they could get their hands on.
So, TL:DR: Boudica very likely did not have 200,000 warriors, and probably not even 100,000, and the Romans were just MUCH better at this.
Edit: It is important to remember that most casualties occured after one army routs from the field. The Romans pursued, cutting down fleeing, panicked people with that hot, furious virtus matched with cool discipline from the officers. Cavalry were exceptionally useful here. Hemmed in by their wagons and the terrain, and hounded by infantry and cavalry, the Britons simply collapsed as a fighting force.
Sources
Primary:
Dio, Lucius Cassius. Roman History, Books 19-22.
Josephus, Flavius (Yosef ben Matityahu). The Jewish War, 6.257-6.258.
Tacitus, Publius Cornelius. Agricola, Book 14.
Tacitus, Publius Cornelius. Annals, Book 14.31 - 14.37
Secondary:
Cowan, Ross. For the Glory of Rome: A History of Warriors and Warfare. Frontline Books: Chicago, Illinois, 2015 edition.
Goldsworthy, Adrian. The Complete Roman Army. Thames and Hudson Publishing: London, U.K., 2005.