how to write powershell script to list the available databases from multiple servers?

You can use this script to find all reachable instances on your network and running your query there:

Import-Module SQLPS -DisableNameChecking 
$servers = [System.Data.Sql.SqlDataSourceEnumerator]::Instance.GetDataSources() 

ForEach ($i in $servers) { 
    $instance = $i.ServerName+"\"+$i.InstanceName
    Invoke-SQLcmd -Server $instance -Database master 'select @@servername as InstanceName,name as DatabaseName,state_desc as DBStatus from sys.databases' | Format-Table       
}

If you need only server name to pass then use $instance = $i.ServerName. Part of code was taken from here long time ago.

EDIT

With writing in CSV file and error catching:

Import-Module SQLPS -DisableNameChecking 
$servers = [System.Data.Sql.SqlDataSourceEnumerator]::Instance.GetDataSources() 
$results = @()
ForEach ($i in $servers) { 
    $instance = $i.ServerName+"\"+$i.InstanceName

    try {
        $sqlres = Invoke-SQLcmd -Server $instance -Database master 'select @@servername as InstanceName,name as DatabaseName,state_desc as DBStatus from sys.databases'

        ForEach($st in $sqlres) {
           $instanceinfo = @{            
                    InstanceName = $st.InstanceName
                    DatabaseName = $st.DatabaseName                 
                    DBStatus     = $st.DBStatus
            } 
            $results += New-Object PSObject -Property $instanceinfo     
        }
    } catch {
        "error when running Invoke-SQLcmd "+$instance
        Write-Host($error)
    }
}

$results | export-csv -Path D:\sql_instances_info.csv -NoTypeInformation

Im not sure what is the problem here. You can put all servers/instances in txt file and iterate:

#array of addresses, this can be fetched from file
$list = "localhost\SQL2014",".\SQL2014","(local)\SQL2014" #MyServer\MyInstance

$list | `
  % { Invoke-Sqlcmd -Server $_ -Database master 'select @@servername as InstanceName,name as DatabaseName,state_desc as DBStatus from sys.databases' } | `
  Format-Table -AutoSize

If those are remote servers without integrated security you would need to pass -UserName and -Password arguments.