First, let's be clear, the reason the rich pay little tax doesn't have much to do with the capital gains tax rate being lower.
Now the reason for the lower rate (at least ostensibly) is that while income is earned at a point in time, capital gains happens over large amounts of time. Therefore often a big part of the gain is inflation. Let's imagine you bought a house for $100k and 20 years later you sell the house for $140k. Over that time inflation has been a steady 2%.
Due to inflation $148k is now worth what $100k was worth 20 years ago. But when you sell you have to pay tax on the $40k profit even though you actually made a loss?
Lower capital gains rates are meant to adjust for this. Basically saying we understand part of the gain is inflation, so let's call it half inflation and half profit and we'll account for this by setting the capital gains rate at half the income tax rate.
Remember companies (that you might have shares in) or yourself as a land lord are (ostensibly) paying tax on profits as you go. Capital gains tax is in addition to this.
This comment is already long enough so I'll leave the conversation on whether this stuff is true in practice as an excercise for the reader, but it at least starts from a sensible place.
At least where I live (not the US), if you're day trading stocks or flipping houses you'll pay income tax not capital gains tax (ostensibly 😆).