Inter-node conflicts arise as a result of sequences of events that could not happen if all the involved transactions happened concurrently on the same node. Because the nodes only exchange changes after transactions commit, each transaction is individually valid on the node it committed on but would not be valid if run on another node that has done other work in the mean time. Since BDR apply essentially replays the transaction on the other nodes, the replay operation can fail if there is a conflict between a transaction being applied and a transaction that was committed on the receiving node.
The reason most conflicts can't happen when all transactions run on a single node is that PostgreSQL has inter-transaction communication mechanisms to prevent it - UNIQUE indexes, SEQUENCEs, row and relation locking, SERIALIZABLE dependency tracking, etc. All of these mechanisms are ways to communicate between transactions to prevent undesirable concurrency issues.
BDR does not have a distributed transaction manager or lock manager. That's part of why it performs well with latency and network partitions. As a result, so transactions on different nodes execute entirely in isolation from each other. Despite the usual perception that "more isolation is good" you actually need to reduce isolation to prevent conflicts.