Why sell stocks when inflation is high?

For any transaction, there must be a buyer and a seller so even when newspapers talk about a "sell-off" or "selling pressure", there are always an equal number of buyers and sellers. Whether the price of stocks are going up or down or staying the same, you've got the same number of buyers and sellers.

If inflation increases and nothing else changes, a stock is intrinsically worth less under basically any valuation model you'd care to choose. If you're trying to determine what a share of Acme Widgets is worth, that's going to depend on the present value of the profits Acme is going to make in the future. In order to determine the present value of a fixed amount of profit a year from now, you need to apply the discount rate. And the discount rate will increase when inflation increases because a dollar a year from now is worth less when inflation is 5% than when it is 1%.

Picking the simplest stock valuation model, the dividend discount model

Fair Price of Stock = (Dividend Amount)/ (Cost of Capital - Growth Rate of Dividend)

where

Cost of Capital = Risk-Free Rate + Beta * Market Risk Premium

When inflation increases, the risk-free rate increases (investors want the government to pay more interest when they borrow money in a 10% inflation environment than in a 1% inflation environment). If nothing else changes-- the dividend remains the same and the growth rate of dividends remain the same (beta and the market risk premium are functionally constants in this discussion)-- then the fair price of the stock declines when inflation increases. If everyone in the market knows the fair price has declined, people will buy and sell at a lower price and the market price of the stock will decline.

Of course, this vastly simplifies the reality of the situation. There are lots of different valuation models that make lots of different assumptions. Different market participants use different models, input different parameters, etc. Some market participants pay attention to valuation models, some don't. When inflation rises, different companies are affected differently. Some companies are stuck absorbing some or all of the increase in prices of their inputs because they're locked into contracts with their customers. Some companies can pass along price increases from their suppliers so their dividends and dividend growth rate would simply increase along with inflation so inflation wouldn't change their prices. Some companies can use an inflationary environment to increase prices and see their growth rate increase faster than inflation.