Your goal is to solve each level with as few intrusions as possible. There are several strategies that can help. First off, if the grid has keys 1, 5, and 7, the possible solutions are going to be 2-4, 6, and 8-9. Your best bet is to place a 3 in one box. If the number turns red, you can eliminate numbers 2-4 as possible solutions for that box.
Try to completely solve one box before moving on to the others; that reduces the pool of numbers you need to guess from. Every number you get correct on the first try reduces your intrusion count by one!
Once you've solved a box, you can drop the same number into that box with no penalty. This can be handy when you're juggling a lot of numbers!
/tmp (temporary) numbers fade out after the timer runs out. If you place another /tmp number in a box that already contains a /tmp number, its timer is reset (however if you place a NON-/tmp number in the box, the timer doesn't reset or go away - the new number will fade out!). Once the timer runs out and the number begins to fade, it will continue to fade out even if you put another number there! Wait until the number has completely faded before putting a new number in the box.
Rotating keys change their solution with another number in the grid. If you already have several numbers solved, watch the key as its timer begins to run out, and remember what number is in the key box. Once it changes, if you see one of your already-solved numbers change color, you'll know that its new solution is the number that used to be in the key box.
Viruses infect neighboring numbers after the timer runs out. Your best bet is to not allow viruses in the grid at all; you can hold a falling number with your finger to keep it from entering the grid or falling off the board. Of course, you only have so many fingers...
If a virus does get into the grid, try to put it in a box with no other numbers around it. Solve the boxes farthest from the virus before solving the adjacent numbers. Once a virus is in the grid, it will spread, changing a neighboring number after the timer runs out. If it has no neighbor, it will reset its timer and wait until it does have a neighbor to infect...
You can also use a vaccine to kill a virus. Vaccines have no effect on non-virus boxes, nor do they persist after they land in a grid box.