[What does it mean to increase into every st to end? does it mean that after every single stitch i add one?
what does it mean to (inc1,K1)? is it the same as above?]
Increasing into every stitch means, as you say, to make a new stitch for every stitch on your needle, ending up with twice the number you started out with.
If you (inc1,K1) you are also increasing one stitch for each stitch on your needle.
There are videos showing various ways to increase or "make " a stitch. One way is to knit into the front and back of a stitch, (kfb) although this doesn't look so good if you are doing stockinette stitch as it leaves a little bump on the front. You can "make" a stitch (m1) in different ways by picking up between or below the stitches on your needle.
Look at the videos and pick one method that makes sense to start out with; you can try other methods later on.
Good luck, I'm sure your child will love her monkey.