Life In 19x19

in one of guo juan's opening training for beginners lectures she presents a question of where white should play, and plays out the response to white's invasion at M-17, saying that white's thickness on the left makes this possible.

i am not very good at fighting, so i don't really get it. i have tried playing it out a bunch of times in different ways but it always ends up some sprawling mess that isn't a noticeable clear cut victory for white

Practical fighting sequences, which can arise from that position are quite manyfold (and cumbersome to post here). They require some fighting strength / reading skill, which you can generally improve by playing games and solving many tsumego. On the other hand, during local fighting it is always good to have some "goal"

Here the goal would be something like "split the b stones and let them each face a large number of w stones in order to get an uneasy feeling". If you successfully manage to split K17 and O17/16, black would have to struggle a lot to save both sides. Either one of both sides dies, or you get many extra free moves, which you can use for solidifying your corner and right side territory. Even if b manages to live somehow in mid-air, you will get many secured territory points by this way.

Here's a try on some sequences. No quarantee on correctness. Wonder whether I got the attachment thing right, download it if it doesn't work otherwise.

Cherryhill, this is something that comes naturally to stronger players. The idea is to use the (very real) threat of the many White stones to gain somewhere. If you think about your basics, you'll remember that corners are valuable because it takes two walls to make territory, the two joining sides making for natural walls of their own.

What then of a wall facing the outside? At your level you might be confused about some joseki where one player takes a small wall facing out. "This is an even result", most say. The first thought that might come to mind is that you'll need two more walls to make the territory the other player made in the corner. Of course this would be inefficient. The first player would win for sure.

So it comes down to making more efficient use of that wall. If you want to make territory out of those stones, there are two basic methods. You either:

A) Kill something with it (2 points for every stone killed versus one for regular territory)
B) Threaten to kill something until you get enough 'free moves' to make territory somewhere else

Guo Juan is demonstrating both approaches here. White's wall is perfect (no defects, all stones connected) and huge, and if Black extends to make a base in that area, then if you play something along the sequences TIM82 has posted, there's a very real chance half the Black group will die. Meanwhile, you're going to be making an amazing amount of points on the right. And the White stone at Q10 means you can skip reinforcing the right side for one move, since it's already there.

Some of the confusion might also stem from the last move from White shown in your diagram. The beautiful thing about thickness is that you can play otherwise unreasonable moves to great effect. Take any of TIM82's sequences and play them out on an empty board. The result will be terrible for White every time. Because of the wall and the stones on the right, however, it's Black who, split up like he is, is in serious danger.

Instead of the kick, the beginner's instinct is to pincer, but in this case the wall is too close and inefficient. Black dives into the corner, gets territory, and White comes out of it with two walls facing each other. They're now unusable and White will have to add at least one more move in that area to keep Black from reducing it.