On free tier, you can send 3 DMs per day to your connections, free. After that, additional DM replies cost 1 Spark each.
The 3-per-day count resets at 00:00 UTC. It applies across all your DM threads combined, not per-thread. So you can send 3 DMs to one person, or one to each of three people; it's all the same daily allowance.
Receiving DMs is always free. You can read incoming messages anytime, in any number, without spending anything. The Sparks/quota only applies to messages you send.
After the first 3 free messages, each additional free-tier DM costs 1 Spark. Premium DMs do not cost Sparks. Both tiers stop at the 200-message-per-UTC-day safety and abuse-prevention cap.
Why the limit on free? Same logic as the connection-request cost: it keeps the network calm. Someone with infinite free DMs becomes a different kind of presence than someone choosing each message deliberately. The 3-per-day limit isn't punitive (3 is plenty for a normal conversational exchange), but it filters out the kind of behavior that hurts everyone: the constant pinging, the multi-pronged messaging campaign, the bot-like behavior.
If you find yourself regularly hitting the limit and buying Spark Packs to extend, Premium ($9/mo) is usually cheaper.