First, I will introduce you to a known phenomenon for CEO salary. As linked, it is demonstrated time and time again that the mandate to disclose CEO pay had a very stark effect on CEO pay overall. CEO pay compressed very quickly, as each CEO was able to see what the others made. Everyone went for the high score, essentially, or at least to an even keel.
However, this does not mean that it is bad to mandate the disclosure of pay, and that's not relevant to what I'm trying to demonstrate here. What this demonstrates is that the business world knows what happens when people are aware of what each other makes. That's why they don't want people discussing it. They will take that benefit for themselves, but you don't get the advantage of knowing your market value relative to others to see if it aligns with what you believe you're worth. They want to deny you the advantage of negotiating wages collectively, with the strength that collective bargaining provides at the negotiating table, by trying to convince you that you are stronger on your own.
That is one reason that employers don't disclose salary. The other one is because of anchor points.
In negotiation, the first offer made establishes what is called an “anchor point.” The anchor point sets the stage for the negotiation. Sometimes, it's good to be the one to establish the anchor point in a negotiation, like if you want to say your widgets are worth a certain amount, you list the price. If you don't list a price, then someone can come in offering any amount they feel like, and that could be far below what you could accept. On the other hand, if you have a maximum that you could accept in a negotiation, such as for a salary, then tipping your hand by establishing that anchor point will tell the other guy in the negotiation exactly what your maximum is, and it will allow them to just take you for everything they can.
So, by not posting a salary, the hiring manager is able to allow the interviewee to set the anchor point, which will very likely be below the hiring manager's maximum. Once you say what you are willing to accept, you cannot negotiate for a better position, because they know you're willing to accept less. They wait for you to establish the anchor point, so they don't have to tip their hand. They can fish until they find someone who accidentally lowballs the salary, and then further negotiate that person to an even lower salary.
Now that you know what an anchor point is, and you know that this is a genuine negotiation, you can understand exactly why they ask you what your previous salary was. They are attempting to set the anchor point at your previous salary. When they tell you not to disclose your salary afterward, they are trying to make sure it sticks.