Parsing the log file using select-String

To get the filenames from the log where the date is for today, you can use this:

$logfile = 'C:\Creds\AzCopyVerbose.log'
$today = Get-Date -UFormat "%Y/%m/%d"
$pattern = '^\[(?<date>\d{4}/\d{2}/\d{2}).*Finished transfer:\s+(?<filename>.*)\s+=>.*$'
Select-String -Path $logfile -Pattern $pattern | Select-Object -Last 20 | ForEach-Object {
    $null = $_.Line -match $pattern
    if ($matches['date'] -eq $today) {
        Write-Host $matches['filename']
    }
    else {
        Write-Host "The date present in the azcopy verbose log is older"
    }
}

Instead of using Select-String, you can also do this which requires to match the pattern only once, so a bit cleaner in my opinion:

$logfile = 'C:\Creds\AzCopyVerbose.log'
$today = Get-Date -UFormat "%Y/%m/%d"
$pattern = '^\[(?<date>\d{4}/\d{2}/\d{2}).*Finished transfer:\s+(?<filename>.*)\s+=>.*$'
Get-Content -Path $logfile | Select-Object -Last 20 | Where-Object { $_ -match $pattern } | ForEach-Object {
    if ($matches['date'] -eq $today) {
        Write-Host $matches['filename']
    }
    else {
        Write-Host "The date present in the azcopy verbose log is older"
    }
}

p.s. I use Get-Date -UFormat "%Y/%m/%d" because (Get-Date).ToString('yyyy/MM/dd') on my Dutch machine outputs 2018-11-10

Regex Details

^                     Assert position at the beginning of the string
\[                    Match the character “[” literally
(?<date>              Match the regular expression below and capture its match into backreference with name “date”
   \d                 Match a single digit 0..9
      {4}             Exactly 4 times
   \/                 Match the character “/” literally
   \d                 Match a single digit 0..9
      {2}             Exactly 2 times
   \/                 Match the character “/” literally
   \d                 Match a single digit 0..9
      {2}             Exactly 2 times
)
.                     Match any single character that is not a line break character
   *                  Between zero and unlimited times, as many times as possible, giving back as needed (greedy)
Finished\ transfer:   Match the characters “Finished transfer:” literally
\s                    Match a single character that is a “whitespace character” (spaces, tabs, line breaks, etc.)
   +                  Between one and unlimited times, as many times as possible, giving back as needed (greedy)
(?<filename>          Match the regular expression below and capture its match into backreference with name “filename”
   .                  Match any single character that is not a line break character
      *               Between zero and unlimited times, as many times as possible, giving back as needed (greedy)
)
\s                    Match a single character that is a “whitespace character” (spaces, tabs, line breaks, etc.)
   +                  Between one and unlimited times, as many times as possible, giving back as needed (greedy)
=>                    Match the characters “=>” literally
.                     Match any single character that is not a line break character
   *                  Between zero and unlimited times, as many times as possible, giving back as needed (greedy)
$                     Assert position at the end of the string (or before the line break at the end of the string, if any)

To complement Theo's helpful answer:

It is not obvious, but it is possible to access a Select-String command's [named] capture-group matches in a ForEach-Object script block (no need to repeat matching with -match):

PS> '... File transfer: C:\path\to\file => ...' |
      Select-String '\bFile transfer: (?<file>.+?) =>' | 
        ForEach-Object { $_.Matches[0].Groups['file'].Value }
C:\path\to\file  # Value of named capture group 'file' 

• $_.Matches is the collection of matches for the current input line; unless -AllMatches was specified, there's only one entry, with index 0.

• .Groups accesses the collection of capture-group matches (with the entry at index 0 containing the overall match).

• ['file'] accesses the match for named capture group file, but note that at index-based access works equally (for unnamed capture groups), starting with index 1; that is, $_.Matches[0].Groups[1].Value in the command above would have yielded the same result.

In terms of data types, Select-String emits [Microsoft.PowerShell.Commands.MatchInfo] instances, whose .Matches property is an array of [System.Text.RegularExpressions.Match] instances.

Below is an approach using regex for date.

$var = Select-String -Path "C:\Creds\AzCopyVerbose.log" -Pattern 'Finished Transfer' | 
Select-Object -Last 20 | ForEach-Object -Process {
      $_.Line -match '\d{4}/\d{2}/\d{2}' | Out-Null
      [string]$Match = $Matches.Values
      if ([Datetime]$Match -eq (Get-Date).ToString('yyyy/MM/dd'))
      {
        $_.Line. // There is no method which I'm aware of need help in this if statement
      }
      Else {Write-Host "The date present in the azcopy verbose log is older"}
    }