Try this:
echo "Enter Current Date (YYMMDD):"
read date
current_date=`date +"%y%m%d"`
if [ "$date" = "$current_date" ]; then
        echo "match"
   else
        echo "no match"
fi
Change is on this line :
current_date=`date +"%y%m%d"`
Reason:  In your code, you are using current_date= date +"%y%m%d". Though your intention was to use the date command to generate the current date, this line behaves differently (becomes equivalent to current_date= ; date +"%y%m%d"; and results in current_date being empty. You can echo $current_date to verify this. With the correction provided, we are executing the date command and saving the output to current_date variable. Hope it is clear now. I don't know how to explain in a better way... :-)