free from the love of money: not ruled by the desire for wealth
The word is built as the flat denial of another word, φιλάργυρος "money-loving." It attaches the little prefix that means "not" to the front of it, so at its simplest it labels a person who is not a lover of money. It names an inner disposition, a heart that is not pulled by wealth, rather than merely the outward fact that someone has not stolen anything.
In the wider Greek world this was not vague religious language. Cities carved it into stone to honor officials, generals, and doctors who could be trusted with public funds, and moralists listed it among the marks of an admirable leader. It was, in effect, the vocabulary of financial trustworthiness: proof that a person entrusted with other people's money had not enriched himself at their expense. Older reference works wrongly thought the word was coined by the early church, but inscriptions and everyday documents show it was already in common use centuries earlier.
The New Testament borrows that public virtue and puts it to two uses. In 1 Timothy 3:3 it is a requirement for a church overseer, holding a Christian leader to the same standard a Greek city expected of its magistrates. In Hebrews 13:5 it is widened into a way of life for every believer, paired with being content with what you have (αὐτάρκεια) and anchored not in willpower but in God's promise never to leave or forsake his people. It is the positive counterpart to the money-hunger, φιλαργυρία, that 1 Timothy 6:10 calls a root of all kinds of evil.