You can go into the League client (must be logged into an account of the same region) then go to profile and type their account name into the summoner search near the top right of the profile then hit enter. After that you can go to their match history and download any ranked games in their last 20 games played.

When you download a replay from their profile it will locally save a .ROFL file which is the replay file. You can then play the replay from the profile or alternatively manually play the .ROFL file.

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If you want to manually play a .ROFL file here's a guide:

The .ROFL files are stored by default in C:\Users\(YOUR USERNAME)\Documents\League of Legends\Replays

To manually play a .ROFL file you have to go to your League of Legends install path and find the current version of the client.

The default path is C:\Riot Games\League of Legends\RADS\solutions\lol_game_client_sln\releases\0.0.1.249\deploy

The "0.0.1.249" part will change depending on patch, this is just a reference to the current patch(8.24). Move your desired .ROFL file to that folder then drag it onto League of Legends.exe to open it.

Note: You can only play replays from the current patch, if you wish to play replays from previous patches you must have already saved a copy of the game's files from that patch.