phd's helpful answer offers the best solution for the case at hand.
As for why ... | Out-Null wasn't effective:
Out-Null suppresses only stdout output from external programs such as git, but not stderr output.
git, like many CLIs (console / terminal programs), uses the stderr stream not just to report errors, but also for status information - basically, anything that's not data.
To suppress both stdout and stderr output, use *> $null:
git clean -fd *> $null
Note: *> $null suppresses all output streams; while external programs only have 2 (stdout and stderr), applying *>$null to a PowerShell-native command silences all 6 output streams.
See about_Redirection for more information.
Optional reading: Selective stream redirection from external programs:
Building on feedback from nmbell:
- >$null(or- 1>$null) can be used to suppress stdout output selectively, which is effectively the same as- | Out-Null.
 
- 2>$nullcan be used to suppress stderr output selectively.
 
- *>$null, as discussed above, silences both (all) streams.
 
Of course, instead of $null for suppressing output, the redirection target may also may be a file (name or path).
Note:
- PowerShell processes output from external programs line by line in its pipeline, and if the output is captured in a variable (- $out = ...) and comprises 2 or more lines, it is stored as an array (- [object[]]) of lines (strings).
 
- PowerShell only ever "speaks text" (uses strings) with external programs, both when sending and receiving data, which means that character-encoding issues may come into play. 
- See this answer for more information on both these aspects. 
Scenarios with examples:
Setup:
# Construct a platform-appropriate command, stored in a script block ({ ... }) 
# that calls an external program (the platform-native shell) that outputs
# 1 line of stdout and 1 line of stderr output each, and can later be 
# invoked with `&`, the call operator.
$externalCmd = if ($env:OS -eq 'Windows_NT') {     # Windows
                 { cmd /c 'echo out & echo err >&2' } 
               } else {                            # Unix (macOS, Linux)
                 { sh -c 'echo out; echo err >&2' } 
               }
Capture stdout, pass stderr through:
PS> $captured = & $externalCmd; "Captured: $captured"
err            # Stderr output was *passed through*
Captured: out  # Captured stdout output.
Capture stdout, suppress stderr output, with 2>$null:
PS> $captured = & $externalCmd 2>$null; "Captured: $captured"
Captured: out  # Captured stdout output - stderr output was suppressed.
Capture both stdout and stderr, with *>&1:
PS> $captured = & $externalCmd *>&1 | % ToString; "Captured: $captured"
Captured: out err  # *Combined* stdout and stderr output.
Note:
- % ToStringis short for- ForEach-Object ToString, which calls the- .ToString()method on each output object, which ensures that the- System.Management.Automation.ErrorRecordinstances that PowerShell wraps stderr lines in are converted back to strings.
- $capturedreceives a 2-element array (- [object[]]) of lines - containing the stdout and stderr line as elements, respectively; it is PowerShell's string interpolation that turns them into a single-line, space-separated string in this case.
Capture only stderr, suppress stdout:
PS> $captured = 
      & $externalCmd *>&1 | 
        ? { $_ -is [System.Management.Automation.ErrorRecord] } | 
          % ToString; "Captured: $captured"
Captured: err  # Captured stderr output *only*.
Note:
- ? { $_ -is [System.Management.Automation.ErrorRecord] }is short for
 - Where-Object { $_ -is [System.Management.Automation.ErrorRecord] }, which passes only stderr lines - identifiable via the wrapper type being tested for - through, and- % ToStringagain converts them back to strings.
 
- This technique is neither obvious nor convenient; GitHub suggestion #4332 proposes a syntax such as - 2> variable:stderrto support redirecting streams to variables, such as- $stderrin this case.