Finding the memories in The Legend of Zelda: Breath of the Wild is a lot of fun, and really fleshes out the story. Near the beginning of the game, Impa will give you twelve photos and ask you to find these locations to trigger Link's memories of what happened 100 years earlier. A lot of these memories – marked by glowing spots that you can stand in to trigger the cutscenes – are likely to present themselves to you during your travels.
One memory, though, is harder to get to than the others – it's located on the grounds of Hyrule Castle, the game's hardest area. But if you know where you're going and how to get there, it's possible to retrieve this memory with minimal resistance.
To do this, you're going to want a Korok leaf (if you don't have one, try cutting down some trees, or there's a guaranteed one outside a shrine near Eventide Island). I did this with three stamina upgrades and a few stamina-restoring food items.
There are plenty of ways into Hyrule Castle, but for the best approach, warp to Ridgeland Tower and start flying towards the castle.
You'll probably land well before the castle, but keep heading in this direction. This may not seem like the best path, but you won't encounter many enemies on this route, and although the castle is surrounded by water there's an easy way in.
Just off to the west of the castle, by the water, is a raft. This is where you want to use that Korok leaf to propel it forward.
Push yourself to the wall, and then climb up it. This will require quite a bit of stamina, but you've taken down the Rito Divine Beast boss and earned Revali's Gale you have nothing to worry about – it'll take you almost all the way up.
Depending on what part of the cliff you scale here, a Guardian might notice you. Don't worry too much, though - it's easy enough to run to cover, and this one can't move. Head for the wall in front of you and climb.
This is a good time to orient yourself. This is where you're going – a spire just off to the west of the main castle, with a bridge running off it. That bridge is where the memory is located, and you should be able to see the spire from where you are. Observe your map and you should be able to see where you are in relation to it.
The spire is pictured above. Once you've spotted it, start heading that way - it's not far, and if it's not raining you should be able to reach the spire and start climbing within a minute or so. Having done this a few times, I have never encountered any serious challenge going in this direction – in fact, I've never had to fight anything on the way here, and even the patrolling flying Guardian-thing has never given me grief. There are a few things to watch out for, though.
Once you hit the tower, start climbing. There will be a point for you to stand on and reclaim stamina about halfway through, so the climb isn't too hard.
If you start to attract attention from surrounding Guardians, there's an easy solution – shimmy around the spire until you're on the wall facing outwards from the castle. Nothing can spot you there.
In fact, if you climb straight up this wall facing outwards, you'll eventually hit a doorway...
... leading into Zelda's study. Her research notes are here – give them a read if you're so inclined (otherwise check out more Zelda guides for a full recount of what's in them). You're right near the memory spot now though head out and you'll find yourself on the bridge from the photo.
Walk out onto the bridge here and into the glowing light to access the memory and shed further light on the plot. This is a particularly good one, so make sure you visit this spot.
Once the cutscene ends, it's up to you if you want to keep exploring the castle or not – in fact, if you check the map, you're not too far from Calamity Ganon. But keep in mind that finding and watching all of the memories unlocks an extended ending for the game, so you may want to back out now to go find the others ... but this path is worth remembering when you return, if you want a fairly hassle-free route to the final boss.