You'll start off in the middle of the map, surrounded by Garon's forces. The objective is to beat the boss but you can't reach it easily without dealing with some minor hassles. The dragon veins are conveniently located beside Garon and only he can use it to his advantage. After your preparations, create a mid-battle save immediately since the enemies here are relentless and can easily take out even your paired units. The sigils scattered in the map will continuously spawn for Garon. There are no notable drops in this mission so aside from EXP farming, you don't necessarily need to prolong this battle.
Don't scatter your forces. The berserkers are units you have to look out for since they can one-hit kill most of your units with low defense so toggle their attack range always to know where can you place your unit safely. Like before, lure your enemies by deliberately placing your defensive/ most powerful units within their range. Also, you don't need to prolong this battle than necessary. All you have to do to is to ensure that you'll be able to regroup and finish off Garon during your turn.
At the end of the 4th turn, Garon will activate the first dragon vein, changing a large chunk of the terrain to the SW of the map to blight, which will allow him to heal a considerable amount of HP each turn. If you didn't have problems defeating Xander earlier and Garon in his normal form, his dragon form isn't that much of a difference. Send your most powerful units and land the finishing blow, provided that your movement range isn't hampered by his minions, which of course you need to take care of first.
After finishing off Garon, watch the ending scenes. Congratulations for finishing the Birthright Route!