The answers by Gabriel Luci and Mathias R. Jessen give good advice on "fuzzy" filtering of AD users.[1]
However, your primary problem is that your ForEach-Object call is not receiving pipeline input, because you haven't connected it to output from the Import-Csv call.
Secondarily, your -Filter argument { Name -like "'$($_.name)'"} mistakenly applies two layers of quoting and is missing wildcard characters, given that -like compares against the entire field, yet you want substring matching.
Since it's better to avoid the use of script blocks ({ ... }) as -Filter arguments, use a string:
"Name -like `"*$($_.name)*`"" # Note the enclosing '*' to match substrings
Note that I've used embedded " quoting (escaped as `") rather than ' quoting, so as not to break the filter with names that contain ', such as O'Malley.
That said, if, as your question suggests, the names in your CSV data aren't direct substrings of the AD users' .Name property values, the above filter will not be enough, and even the ANR (Ambiguous Name Resolution) technique shown in the linked answers may not be enough.
Thirdly, your Export-Csv call is misplaced: because it is inside the ForEach-Object script block, the same output file gets overwritten in every iteration.
Optional reading: ForEach-Object behavior when not providing pipeline input:
The associated script block is executed once.
$_, the automatic variable that contains the current input object, is $null.
[1] Note that the search term in the LDAP filter may need escaping ; per this article, the characters * ( ) \ NUL require escaping and must be escaped as \<hh>, where <hh> is the two-digit hex representation of the char's ASCII code (e.g., * must be escaped as \2A):
$escapedName = -join $(foreach ($c in [char[]] $_.name) { if ('*', '\', '(', ')', "`0" -contains $c) { '\' + ([int] $c).ToString('X2') } else { $c } })
Get-ADUser -LDAPFilter "(anr=$escapedName)"
With $_.name containing string "James* (Jimmy) Smith\Smyth`0", $escapedName would evaluate to literal James\2A \28Jimmy\29 Smith\5CSmyth\00