Your grandfather dies. In his will, he leaves you his house, which has been in your family several generation. It's a nice place, better than the crappy apartment you live in, and it's yours now by all rights, so you move in.
When you do, you discover to your delight that you water bill is zero dollars every month. What a nice bonus! Free water from the tap!
One day, when poking around the basement, you discover the secret to this mystery. Apparently one of your ancestors many years ago dug a secret tunnel over to the neighbor's house put a T on their water main, and ran a pipe back to your house. Your water isn't free. Your neighbors have been paying for it the whole time.
In fact, they have even known this and been trying to tell you. But, you know, you were so busy getting settled in and dealing with all the stuff in your own life that their discussion about "water equality" never really registered for you. It's not that you didn't care (you love equality), you just didn't think it had anything to do with you.
So what is your moral position today?
It is not one of guilt. You didn't put that sneaky pipe in. And while, yes, you certainly took some showers with free water, at the time you honestly didn't realize that anyone was paying for it. There was no malice on your part.
You could argue that since it's your neighbors who are suffering from the jacked up water bill, they should be the ones to pay to cut that pipe and remove it. After all, you didn't cause the problem, and you don't have any personal incentive to fix it. You aren't trying to steal their water, it's just the way your plumbing happens to be set up.
At the same time, your neighbors are actually poorer than you, in large part because they have been paying your water bill the whole time. It feels pretty selfish to expect them to foot the plumbing bill to get it fixed.
So I think that you bear a responsibility to fix the plumbing because you now own the house. And you have some moral obligation in the sense that those who are most able to do a thing bear some obligation to their community to do that thing. If we're all in this together, then we give back to society in the ways we best can. Since you can more easily afford to the fix the plumbing (all those months of free water let you save up some cash), you should be the one to do so.
Now, granted, there are certainly some progressives (of all races) who take the history of slavery in a guilt/shame direction. If you're white, you're just supposed to feel bad. We should all be walking around in hairshirts as a penance for the sins of our fathers. There is a real weird Catholic guilt vibe in some progressive circles today.
But I think for most, it's not that. And the most charitable interpretation of people saying that whites today bear responsibility for slavery is just what my example here says: we have some ownership over the institutions that benefited from slavery, and we have a greater capacity to amend that problem, thus responsibility to do that falls on our shoulders.
The most compassionate way to look at this is as an opportunity. What a great thing it is to be in a position to help address one of the most grievious injustices in the United States.