When most people who play chess need to look a few ply ahead they do so by visualizing the board and pieces and them moving those visualized pieces around on that visualized board. They pretty much do with the visualized board what they would do if they had access to a physical or computer analysis board that they could actually move pieces around on.
I once wondered if top players do it that way too, or if maybe the see the position in some more abstract way like a graph with pieces as vertices and colored directed edges encoding relationships such as "The rook is attacking that night" and "that knight is defended by that bishop" and moves are then operations that shift edges.
I asked GM Nakamura about it on an AMA he did on Reddit, and he said he sees the board just like nearly everybody else does.