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For example, let's say we have a uniform sales tax of 5%. A poor person with an income of $20k will spend almost all of that $20k during the year, paying 5% of her salary in taxes. (She might even borrow money and spend more than her income, meaning relative to her income the rate increases to more than 5%.)

Say for the sake of argument unpoor me makes $40k and I spend $30k, saving the remaining $10k. I pay $1.5k in taxes, more in absolute terms but less as a percentage of income (3.75%), which is the definition of a regressive tax.

There are other ways sales taxes tend to favor the unpoor. For example, a poor person will typically buy almost all goods locally and thus be subject to tax on all of them. This is not necessarily true of non-poor people -- more than half of my monthly purchases are tax-exempt because of where I make the purchase. (For example, when I buy software from American companies, neither Japan nor my locality nor most US states tax me.)

There are tweaks you can make to sales taxes which make them marginally "fairer" to the poor: exempting food and clothing, for example, which the poor tend to spend more on as a percentage of income than the rich. However, these tweaks get very complicated very fast (a lot of governments actually have a policy which sounds like "Exempting food sales but taxing prepared food sold for immediate consumption unless it is take-out except taxing hot take-out." I wish I was joking. My sympathies in advance to anyone who writes POS software for French bistros.) These exemptions also tend to cost the state a lot of money, because non-poor people spend lots of money on food, clothing, etc.