The right way is:
db.users.find({awards: {$elemMatch: {award:'National Medal', year:1975}}})
$elemMatch
allows you to match more than one component within the same array element.
Without $elemMatch
mongo will look for users with National Medal in some year and some award in 1975s, but not for users with National Medal in 1975.
See MongoDB $elemMatch Documentation for more info. See Read Operations Documentation for more information about querying documents with arrays.