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This isn't the only reason, but one of the current reasons is that Donald Trump has not divested himself of any of his business holdings before assuming office. He has continued to profit off of these businesses throughout his term in office. The emoluments clause exists for a reason. There is a great deal of evidence of foreign powers paying money to Trump hotels in order to curry favor with him.

In short, his tax returns give people a way to examine whether and how he is being bribed. It's not the whole story, but it is a critical part of the puzzle.

There's also the fact that the public wishes to have someone honest enough to reveal their tax forms to serve as president. It's called transparency. Trump promised that he would during his campaign, and then betrayed his own promise to do so. That makes him a liar and a cheat.

As to who cares and wants to see them released, polls say something over half the voters do, e.g. https://morningconsult.com/2019/04/10/most-voters-support-democrats-quest-for-trumps-tax-returns/ (Numbers of course vary by the pollsters and dates, but AFAIK it's always been a fairly large share of the electorate.)

Of course we don't know why all these people would like to see them released. There are probably many different reasons, ranging from "It's traditional, all the other Presidents/presidential candidates always have." to "It'll show that he's committing tax fraud or other crimes, lying about his wealth, and so on."

As for why he's making a big deal out of not releasing them, there seem to be only two possible reasons. Either he knows that they really will prove that he's committing fraud and/or lying about his wealth, or it's simply a matter of ego.

It would certainly be of interest to the Attorney General of the State of New York, who is currently suing Trump over the legality of his charity organizations.

The lawsuit alleges that Trump has been earmarking charitable donations that have actually been used to benefit Trump's personal, business, and political interests.

The lawsuit, filed in State Supreme Court in Manhattan, culminated a nearly two-year investigation of Mr. Trump’s charity, which became a subject of scrutiny during and after the 2016 presidential campaign. While such foundations are supposed to be devoted to charitable activities, the petition asserts that Mr. Trump’s was often improperly used to settle legal claims against his various businesses, even spending $10,000 on a portrait of Mr. Trump that was hung at one of his golf clubs.

Charity organizations are typically tax exempt - but they have to actually be charity organizations legally in order to maintain this tax exempt status.

By December, the Trump Foundation agreed to dissolve under court supervision due to this lawsuit.

The foundation was accused by the attorney general, Barbara Underwood, of “functioning as little more than a checkbook to serve Mr. Trump’s business and political interests,” and of engaging in “a shocking pattern of illegality” that included unlawfully coordinating with Mr. Trump’s 2016 presidential campaign.

In addition to shuttering the charity, her office has pursued a lawsuit that could bar President Trump and his three oldest children from the boards of other New York charities, as well as force the payment of millions in restitution and penalties.

In the past, Trump had engaged in a scheme with jewelry stores to evade New York sales taxes:

Trump would go into the store with his wife, his girlfriend, his...whatever (to use his vernacular). He would then buy her an expensive necklace or wristwatch. Normally, such a transaction would face the New York city and state sales tax, which would be pretty high on luxury jewelry.

In an illegal attempt to evade the tax, Trump "asked" the store to instead ship the jewelry to an out of state location, where no New York sales tax could be collected. In fact, the store would merely send an empty jewelry box to the location, while Trump and his lady friends walked out the door with the jewelry that very day.

The state and city tax collectors eventually caught onto this scheme, and Trump promptly testified against his erstwhile tax evasion colluding partners at the jewelry store in order to save his own skin.

In short - it is of legal interest to the prosecuting team of New York State suing him for tax evasion as well as showing proven evidence of a historical pattern of tax fraud.

This answer presumes that the returns contain nothing politically damaging to Trump.

In this case, refusing ignites his opposition and gets them focused on getting the returns publicized. They begin to speculate about what is in them, and spend a whole lot of time an energy on them.

Then when he decides that he has wasted enough of his opposition's time and energy, he releases them, and it's a nothingburger, and he catches his opposition flat-footed.

It's an added bonus if his 2020 opponent stakes his/her credibility on there being something illicit in those returns, and loses face when this turns out not to be the case.

(The fact that the 2016 election cycle went by without someone in the IRS anonymously leaking them leads me to believe that there is nothing at all damaging to Trump in his returns.)