When economists talk about markets, they treat them as if every market is an auction.
Think about it this in terms of eBay. If there's something up for auction there but there are a million of them, there's no shortage of it and people don't compete against each other to buy it -- there are plenty for everyone. But then compare it with something rare -- the bids go up and up because there is, in essence, a shortage.
Another way to think about it is in terms of great NBA or NFL players. There's always a shortage of those so the teams will pay them huge amounts when they become free agents.
It doesn't work exactly like that in the market for regular stuff, but it's the same idea. When firms see that their product sells out all the time, they'll raise the price. The consumers aren't literally bidding the price up, but the effect is the same.
I am not sure if it is appropriate to take the phrase "bid up the price" too literally except in markets where the selling and buying is done through the process of auction, as it is done on eBay.
A typical competitive market is different from auction in many ways. In auction each product on sale is unique, and there is only one supplier. Also for each product there is only one buyer. Though many people may bid for the product, finally only the highest bidder gets to purchase the product. In contrast a normal market has is characterized by a standard product with many buyers and many suppliers. Also it is not as if a customer will totally stop buying a product because of increase in price. It is more likely that a customer will reduce the quantity purchased.
So I believe the phrase 'bid up the price" simply implies that the customer are willing to or offer to pay higher prices for a commodity they find is in short supply. But at the same time they may also reduce the quantity they purchase. When the sellers find out that customers are thus willing to pay higher price, and suppliers in any case do not have enough quantity available to sell the whole quantity demanded at existing prices, they may increase their prices.