Something done assiduously is done with care and dedication. If you assiduously count your lemonade stand earnings, you tally up the dimes and quarters very carefully, possibly counting them twice to make sure you get it right.
Diligence, care, thoroughness, precision — all of these describe what's needed to do something assiduously. It comes from the Latin word assiduus, which means "busy, incessant, continual, or constant." The people who shelf books at your local library do so assiduously — they take great care to put the books exactly where they belong in the stacks so that others can find them.