Why do so many Democrats favor the elimination of the filibuster?

Because the filibuster has been a huge benefit to the political right over the years. Yes, it has also occasionally benefited the left, too. But only occasionally. Consider the history:

(1/x)

In the 1840s (before the term existed), Senator John C. Calhoun used the technique to protect slavery.

Over the next century, Southern Democrats repeatedly used the filibuster to prevent Black Americans from voting and to defeat anti-lynching bills.

(2/x)

From the 1950s through the 1990s, Senate Republicans, working with some conservative Democrats, blocked the passage of laws that would have helped labor unions organize workers.

(3/x)

Over the past two decades, the filibuster has enabled Republicans to defeat a long list of progressive bills — on climate change, oil subsidies, campaign finance, Wall Street regulation, corporate offshoring, gun control, immigration, gender pay equality and Medicare expansion.

The list of progressive uses of the filibuster is far shorter: some blocked judges, the defeat of some abortion restrictions and an estate-tax cut.

On balance, the filibuster helps Republicans. Its elimination would almost certainly help Democrats.

This makes sense, too. Consider the definitions of the words “conservative” and “progressive.”

As @AJentleson says, “The filibuster is a tool to preserve the status quo and makes it harder to make change.”