What does "burned through $300 million" mean?

Burn rate describes the rate at which a new company spends its capital to finance overhead before generating positive cash flow (negative cash flow). So burned through means they accomplished it and ran out of money.

This answer is in response to OP's comment (now edited into the post):

Was it angel investors' money, and they spent it without any profit? Or was it the value of the company at the time of the dot com bubble, and its market value lost so much in value after the bursting of the bubble?

You can't (directly) spend the "value of your company" (the amount people are willing to pay for a share of stock times the number of outstanding shares).

You can only spend cash.

As noted in my comments elsewhere, Pets.com is a symbol of 1990s hubris in the dotcom era, when everyone assumed they could build a model that started with heavily discounted merchandise to lure in new customers and then get them to spend more on profitable items.

The fundamental problem with this model was the inability of the companies to understand that cheaper prices were always just a quick search and mouse click away, so there was no longer the same kind of brand loyalty that had been counted on for years in the brick-and-mortar retail trade. There, people weren't going to get in their car and drive across town to save a few dollars on an item - the extra effort didn't justify it. But with the explosion of online shopping, better prices could be hand with a few simple keystrokes.

Amazon.com almost fell victim to this, when it started out selling books at near-wholesale price assuming people would return to buy other items at higher markups. Fortunately, Jeff Bezos had both the foresight (and investor capital) to pivot his business model into other areas where he could get improved margins and survive until he could become profitable.

Companies like Pets.com, on the other hand, stupidly spent money on things such as ridiculously-overpriced Super Bowl ads which may have been cute, funny, and discussions around the water cooler at work for a few days but did nothing to generate revenues anything close to what was spent.

Since nobody had a real understanding what the internet shopping world would look like in its early days, investors were willing to fund virtually anything with ".com" at the end of it, assuming there was no real cost in setting up an online presence. Once it became clear what the real model was actually turning out to be, the cash spigots suddenly turned off, and companies like Pets.com, which assumed an endless supply of investor capital, quickly ran out of money and died.

So, in the end, Pets.com raised money from a combination of angel investors, bankers, and its IPO, and then burned through it in less than a year, never achieving anything other than becoming a footnote in the history of the internet.