So it doesn't seem to be broadcasting the transaction directly to the blockchain but pass through some kind of protected server.
"The blockchain" isn't a single entity which receives transactions from the whole world to one single place/server/node. The bitcoin network is created from thousands of full nodes interacting with each other. When you want to broadcast a transaction, you have to send it to a node, which will then relay that transaction to other nodes, which will each relay it to more nodes, and so on, until it has propagated through the entire network. If you do not run your own node which can broadcast transactions, then you have to send them to someone else's node to be broadcast.
If they use servers, how many are they? 2-3 or hundreds of them?
Depends on the wallet software you are using to broadcast your transaction. Ledger probably run a couple of servers for users to connect to via Ledger Live. Electrum has over 100 public servers you can choose from.