Why do some think the US is more republic than a democracy?

A Democracy is "Rule by all People", where each person has direct input on every law.

A Republic is "Rule by a Representative Group", where only a few people have direct input on every law.

The US is therefore closer to a Republic than a true Democracy. However, it is not a "true Republic". The best description of our government type is Democratic Republic. All people vote to elect the representative group, who are therefore inclined to cast their votes according to the majority desire of the masses they represent, or in the manner they feel best benefits those people.