Everybody pays VAT. As a business you charge VAT for everything you sell and pay that out to the tax authority, and you get VAT back for anything you buy. As a consumer it's just an item on the bill.
Say the VAT rate is 20%. Now if you buy something for $100, install it and charge $100 to your customer, you get back $20 from the tax authority and pay them $20, so if billing cycles align no money actually flows to or from the tax authority. But if you add value, say by buying $100 in parts, assemble them and sell the assembly for $150, you get back $20 for parts purchased but collect $30 for the sale, creating a net flow of $10 to the tax authority.
If everything happens under the same tax authority this nuance doesn't matter, in total there's always a $30 tax on a $150 part, no matter how complex the supply chain. But if more countries are involved the difference matters: if a company in Poland makes parts worth $100 and a company in Germany assembles them and sells them in the German market for $150, that's $20 in taxes for Poland and $10 for Germany. With a sales tax that's only collected when selling to a consumer it would have been $30 for Germany and $0 for Poland.
The Polish company invoices the German company under the "reverse charge" regime. The German company treats the parts as if they were supplied by another Germany company, charging itself German VAT and refunding itself an equal amount of German VAT. There's nothing collected in Poland.
Reverse charge only moves vat collection from seller to buyer, so seller doesn't have to deal with 27 tax agencies and buyer only deals with their tax agency.
Nothing collected directly doesn't necessarily mean nothing collected at all. The turnover subject to said regime still needs to be reported, in higher detail if significant. Although I have no idea what exactly happens with such information, it's not hard to imagine it being used to settle the bill between countries.
It's actually less paperwork, because you don't have two companies filing separate VAT rates with two tax authorities. On the surface, the reverse charge sounds complex but it actually simplifies things for the importer.
That's not how VAT works in EU, when I'm buying stuff as business expenses from EU I need a reverse charge bill with my VAT ID and no VAT is applied on purchase. If I buy locally then I subtract the input when I'm paying my VAT or request a return if it's more than I owe in VAT (which happens when you are selling to other countries).
Say the VAT rate is 20%. Now if you buy something for $100, install it and charge $100 to your customer, you get back $20 from the tax authority and pay them $20, so if billing cycles align no money actually flows to or from the tax authority. But if you add value, say by buying $100 in parts, assemble them and sell the assembly for $150, you get back $20 for parts purchased but collect $30 for the sale, creating a net flow of $10 to the tax authority.
If everything happens under the same tax authority this nuance doesn't matter, in total there's always a $30 tax on a $150 part, no matter how complex the supply chain. But if more countries are involved the difference matters: if a company in Poland makes parts worth $100 and a company in Germany assembles them and sells them in the German market for $150, that's $20 in taxes for Poland and $10 for Germany. With a sales tax that's only collected when selling to a consumer it would have been $30 for Germany and $0 for Poland.