First things first. MLS transfer rules are often made up on the fly depending on the situation (See Jermain Jones' move to Chicago/NE) so what the rules say and what actually happens isn't always close
When players move to the US the club pays the bulk of transfer fee depending on which type of player they are (based on their salary level mainly). Polster isn't a Designated Player iirc so NE Revs will have paid most his fee . If it's a DP, the league pays most of his fee and most of his wages.
So far, so easy. But it gets a bit more complicated when it comes to "(i) select U.S. Men’s National Team players, (ii) elite youth U.S. National Team players, or (iii) former MLS players returning to MLS after joining a non-MLS club for a transfer fee greater than $500,000." If they tick any of those boxes, there in a pecking order over who gets to sign him first with teams finishing at the bottom getting first pick - Polster didn't meet the bar so NE Revs didnt need to worry about losing out in on the allocation spot.
It gets complicated more by teams being able to trade up in the allocation order if there is a player they are desperate to get but they won't get first dibs and another club is interested.
There's another wee quirk too it as well. Deals done within the league don't actually use 'real' money. They use various MLS allocated funds which don't actually exist asproper money (if that makes sense). They can be used to make trades for players or draft allocations and buying down a players impact on the salary cap. (GAM - General Allocation Money; TAM - Targeted Allocation Money)
There's one last wee bit. If a playe has played in MLS and moved out of the league (like Polster) his previous club may retain his player rights - there's a whole set of rules on that too - and in Polster's case NE Revs have paid Chicago Fire £200,000 in GAM over 2020 and 2021 to buy the rights to sign him.