I think most people argue that income inequality is bad on two accounts.
At the upper end, they argue based on a kind of labor theory of income. They ask, if a certain CEO makes 1000x the income of the average worker, is their work really 1000x as difficult, or 1000x as laborious? The answer is obviously no, but that's not the way our economy works. You could ask the same of movie stars or professional athletes. I don't think this is a useful argument. People at this income level get paid what they can negotiate.
At the other end, they argue that it isn't right for some to be so desperately poor. That's why raising the floor of income (perhaps by a Universal Basic Income) is the other part of the argument. To this I'm much more sympathetic. I live in Los Angeles and don't have to go very far in any direction to find a tent city. People are hurting and they need help.
In my opinion, it's not income inequality that is the real problem, but wealth concentration. The concentration of wealth into fewer hands is bad for the economy. If there is less wealth for most people, then there are less purchasers for an economy's output. It's a deflationary scenario where less available money means businesses have to lower prices to sell, making profits smaller and debts harder to pay off. Bill Gates is only going to buy so many TVs, cars, and houses. Doubling his wealth is not going to change his spending habits. If that amount of wealth was placed in the hands of a thousand people, then there would be a thousand new customers for TVs, cars, and houses. This more distributed kind of customer base can sustain an economy.
From this perspective, extreme income inequality is bad because it leads to catastrophic wealth concentration. The small number of very rich can only be customers to a small number of luxury businesses. Every other business relies on the existence of a much larger customer base that can actually afford their wares. If wealth is too concentrated, there's not enough money in enough hands for most businesses to operate. Businesses suffer and lay off their employees, leading to greater unemployment, leading to even fewer customers, leading to worse business, and on down the vicious cycle.