You can get this far with relatively simple arguments based on a single clue at a time:
Then, there's a useful pattern to notice:
If two 1s in the middle of the grid are diagonally adjacent, they cannot be directly connected. That would create a loop around both of them.
This gives us a bit more:
That deduction can actually be extended:
You can't have any "1-2-2-2-2-2-2-1" chains that don't touch the edge, with any number of 2s in the middle, because then you would draw a loop around all of them!
This lets you draw a slash on any 2 that has three 1s diagonally adjacent to it: the fourth edge must be used, because if it wasn't, you'd have a 1-2-1 chain and a loop.
And now it's time for a big deduction:
The lines marked in red must connect to the edge of the grid somehow, because they cannot form a loop.
The yellow-highlighted 1 clues block them off -- they cannot traverse those clues on their path to the edge of the grid. This determines their escape routes:
And now we can finish off the puzzle, with just single-vertex deductions. The solution is below: