grep "$(date +"%b %d")" *files* 

should be better.

See man date

grep "$(date +"%b %d")" *files* 

should be better.

If you leave out the double quotes, then the result of the command substitution $(date +"%b %d") is split into separate words. Always use double quotes around variable substitutions "$variable" and command substitutions "$(somecommand)" unless you know why you need to leave them out.