tl;dr
# Note the nested quoting. CAVEAT: May break in the future.
robocopy.exe | find.exe '"Started"'    
# Alternative. CAVEAT: doesn't support *variable references* after --%
robocopy.exe | find.exe --% "Started"
# *If available*, use PowerShell's equivalent of an external program.
# In lieu of `findstr.exe`, you can use Select-String (whose built-in alias is scs):
# Note: Outputs are *objects* describing the matching lines.
#       To get just the lines, pipe to | % ToString 
#       or - in PowerShell 7+ _ use -Raw
robocopy.exe | sls Started
For an explanation, read on.
PowerShell does support piping to and from external programs.
The problem here is one of parameter parsing and passing: find.exe has the curious requirement that its search term must be enclosed in literal double quotes.
In cmd.exe, simple double-quoting is sufficient: find.exe "Started"
By contrast, PowerShell by default pre-parses parameters before passing them on and strips enclosing quotes if the verbatim argument value doesn't contain spaces, so that find.exe sees only Started, without the double quotes, resulting in an error.
There are three ways to solve this:
- PS v3+ (only an option if your parameters are only literals and/or environment variables): - --%, the stop-parsing symbol, tells PowerShell to pass the rest of the command line as-is to the target program (reference environment variables, if any, cmd-style (- %<var>%)):
 - robocopy.exe | find.exe --% "Started"
 
- PS v2 too, or if you need to use PowerShell variables in the parameters: apply an outer layer of PowerShell quoting (PowerShell will strip the single quotes and pass the contents of the string as-is to - find.exe, with enclosing double quotes intact):
 - robocopy.exe | find.exe '"Started"'
 - 
- Caveat: It is only due to broken behavior that this technique works. If this behavior gets fixed (the fix may require opt-in), the above won't work anymore, because PowerShell would then pass ""Started""behind the scenes, which breaks the call - see this answer for more information.
 
- If an analogous PowerShell command is available, use it, which avoids all quoting problems. In this case, the - Select-Stringcmdlet, PowerShell's more powershell analog to- findstr.execan be used, as shown above.