This checkmate is much more a process than a resulting pattern. The key idea is that the Queen and Rook are exerting parallel fences of power as if they were laying the rails of a train track. Therefore, the opposing King is caught between these two fences.

The attacking Queen and a Rook keep shifting positions, from one snapshot to the other, all the while pushing the opposing King again and again, until the checkmate is made.

So, what are these snapshots in detail? In one snapshot the Queen and Rook are across from one another and the Queen makes check (the Queen and Rook are on the same railroad tie-beam and they are giving mutual protection):

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This position uses the Battering Ram or Barrage tactic.

In another snapshot, the Rook is on the diagonal from the Queen when it makes check (only the Rook is protected, but the Queen is out of reach by the enemy King):

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Note that either snapshot could be the starting point for the process of the Railroad Mate. Some players visualize the process as carrying themselves across monkey bars, swinging from the position of both hands on the same bar to the position with one hand on the next bar, then forward to a new position with both hands on the next bar. Another group of players see the capital letter N (or its mirror-image) being repeated. Still another group sees a rather stilted use of scissors, being closed in one snapshot and opened on only one side in the other snapshot-- 'walking' the scissors across the board.

Of course, you may use your own imagination or forego taking any simulation and enjoy the process as it is.

However you go, do note that the alternation of the snapshots could proceed either along a horizontal path (the default for the diagrams above) or along a vertical path.

The vertical Railroad Mate may be considered by some to be a simplified King Hunt. Korolikov composed the following White to Move and Mate problem that uses helpers to produce a vertical Railroad Mate with Rook and Bishop:

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No, the solution is not given here, but eight moves do the job ;-) It uses two defensive X-Ray tactics that each make a check as well, then ends with an X-Ray Mate that has followed the Railroad Mate process. (If you must have the solution right now, then see Composition #94 in CHESSBOARD MAGIC <1960> by Irving Chernev.)

An interesting shortcut of the Railroad Mate process is to keep the opposing King locked up between Queen and Rook, then Promote to a major piece and bring it for checkmate between the restricting Queen and Rook. (This maneuver is a safe way for less-experienced chess players to use three Rooks or three Queens to make checkmate!)

Staunton vs NN, 1850, 25 moves, 1-0
(bad chessgames.com link) Here, the ending looks like a Railroad Mate, vertically (Snapshot 1) Process of getting to the mate is not the same, though.

There is a Railroad Mate after a long continuation by Stockfish 17 for this game: M Umansky vs M Blokh, 1983 Vertical form (Snapshot 1) but with a Pawn on the team to shorten the process considerably.

In this game, the Railroad Mate acts as the shortcut, after a long continuation R Ruck vs M Hofbauer, 1995 Vertical process was utilized
(One of two lines given by computer analysis)

This game shows the Railroad Mate threat
Smagin vs V Bukhtin, 1982 (Snapshot 2) winning at the last move, triggering a resignation:

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It is an obvious 'Mate, Don't Take!' situation-- do not capture the Black Queen after 34. ... Kxh2. Instead, play 35. Qf2 Kh3 36. Rh1#

There is another game collection, Railroaded into Kill Box or Triangle Mate, which has been compiled by fredthebear, but he makes qualifications on his list. I made my collection independently and before I saw his. So, our descriptions differ. The first snapshot given here is what he called the Triangle Mate (when at the edge and checkmate). The second snapshot given here is what he called the Kill Box, again when it completes the checkmate. Check out what fredthebear did; you should find it useful.

The King Hunt, by John Nunn and W. H. Cozens, should also be cited as a useful resource. That book has several example games that illustrate the Railroad Mate and they are listed here in this game collection.

ChessCoachClark (CCC) originated this game collection in Sept. 2019, but from games and a composition discovered in 2016. He updates it on occasion. This project is a work in progress, culling games from various sources, including several chess training books and personal research. The games are ordered by date (oldest first), not by importance.

Be well.

Be safe.