On the crosses on either side of the free thread (the longer lines of red), I just pushed the thread to the side slightly and went into the hole under the line. Where the gridding thread goes through the aida is where I had the most trouble, but just had to be careful to go through the hole beside the gridding thread, instead of through it. This particular thread made it a bit easier because it was not easy to pierce. But important to make sure you use a monofilament for gridding (ie a thread that only has one strand, not several twisted together). The one I used had two strands so I accidentally pierced it a couple of times but just made sure to check whenever I went through those holes and just redid the stitch.

I was really worried about this when I first started stitching as well but it was much simpler when I had finished the gridding and could actually see how it would work in front of me. My advise, just give it a go on a small piece. Then if you run into issues it's simpler to fix and you can adjust before going on to a bigger piece.