Here is an analogy of how nodes gossips with each others :

In a relay race competition there are 4 runners, Alice, Bob, Carrol & Daniel. Alice has run 10 meters & then handle her baton to Bob. Then Bob has to run another 10 meters & handle his baton to Carrol. Carrol also has to run the same distance & finish the race by handling the baton to Daniel. Think of the baton as the sats.

The routing logic is similar to the way TOR works. Between the sender's node & recipient's node there is/are routing node. Routing node forward sats in return for small fees. The sats are cryptographically locked using HTLC. Basically the payments either 100% success or not happening at all.

You could avoid fees by having direct "channel" towards your recipient ( no intermediary therefore no fee ).

Routing node has to online 24/7. Not everyone should run routing node!. You can just run private/unannounced node & still make payments.Private node does not route txs but can go offline as long as you can trust your peer to not rug you