The chat continues on all linked devices from the point in time that they are linked.
Imagine two people having a face-to-face conversation, then a third person walks up and joins in. The third person doesn’t know what was said before they joined the conversation, but all three continue the conversation from that point on.
Linked devices are like the above example, if two of those people were married and tell each other every conversation they’ve had since their wedding.
I haven’t actually synced a new phone to Signal, does everything just carry over? I assumed you needed to transfer your account from phone to phone, not just link a new device.