Step 2 The 4 clue tells us that there are 4 bridges leaving this island.
But we know that the most bridges we can have to any other island is 2, so we must have 2 in each direction.
Step 3 Similarly the 6 clue tells us that there are 6 bridges leaving this island.
But again we know that the most bridges we can have to any other island is 2, so we must again have 2 in each direction.
Step 4 For each of these islands there is only one island that the single bridge can go to.
Step 5 Each of these islands have only one remaining bridge to find, and it's clear which islands these must join - as the other islands they can reach are either already full or we've used two bridges.
Step 6 The remaining bridges for these islands can only go to one other island.
Step 7 The puzzle is now complete and we can move from any island to any other.