While it is normally preferable to perform OO processing, via ConvertFrom-Csv, in your case plain-text processing enables a faster solution, which matters with large input files.

# Create a hash set for all URNs.
$hs = 
  [System.Collections.Generic.HashSet[string]] (Get-Content -ReadCount 0 urn.txt)

# Read all lines lazily from the CSV file and pass only those
# lines through that are in the hash set.
[IO.File]::ReadLines((Convert-Path records.csv)) | 
  ForEach-Object {
    if ($hs.Contains(($_ -split ',')[0].Trim('"'))) {
      $_
    }
  }

The above directly outputs the results.

• To output to a file, append | Set-Content ..., e.g. | Set-Content filtered.csv; use an -Encoding argument to control the character encoding.

• If you want to write back to the input file (be sure to make a backup copy first), use [IO.File]::ReadAllLines() rather than [IO.File]::ReadLines() or (Get-Content -ReadCount 0 records.csv) (note the parentheses), but note that doing so reads the entire file into an array of lines into memory at once.
  The alternative is to write to a temporary output file first, and then replace the original file with the temporary one.

Note:

• [IO.File]::ReadLines((Convert-Path records.csv)) is used purely for performance reasons, because the PowerShell-idiomatic equivalent, Get-Content records.csv, is regrettably slow – see GitHub issue #7537 for a discussion and a possible future remedy.

• The assumption is that none of the rows in your CSV file span more than one line.