Will the new employer be able to find out about my side business
through my W-2's,
No they will not have access to your W-2 beyond the ones you fill out for them.
They don't see the tax returns you file with the government.
or background checks?
Yes. If they are doing a background check, and they ask about it; you will of course tell them. So they will find out. If they ask and you lie about it you put your job at risk.
Or is it impossible for them to find out unless
I mention it?
If you don't tell them about it they could still find out. If they do a background check they might find reference to it while investigating you online. Of they might discover it if one of your references mentions it.
If the background check has a financial requirement, it is possible they could find reference to it.
Why do they care
It depends on the company. They want to make sure that your main interest regarding work is them. The job might have travel requirements, or the need to work weekends. For some people the side business requirement could limit their ability to be flexible.
They might be concerned that you are trying to grow the business and will not be working with them for very long. They can also be concerned that you will spend work hours, hours they are paying you, instead sneaking in a little work on the side business.
They are also concerned about your business being a competitor, or some other conflict of interest.
To be able to evaluate these issues regarding your specific business they must know about it.
What is the risk
It depends on what you sign, and when you sign it.
Some companies want you declare all outside businesses and projects. They then will claim everything else from that point forward. They will claim that you only started work on X after being exposed to Y at the office.
If they have a requirement to approve all outside income, and you claim that you don't have any. They could then fire you for that omission. That termination could come years after you start.
How likely is this? Not very. It would take a lawyer to give an opinion on how likely it is to fire you, or to claim the business.
But if they ask. You need to tell them, and let them decide.