Its a fisheye lens essentially. The reason why you use the grid is so that you can see where the objects lie in relation to the planes of the picture. I recommend look at some of Escher's stuff and if you can find his pre sketches you will see that you sort of have to be very systematic and mathematical about it, you can do the grid by hand, it just takes alo't of figureing out.
Ultimately what you want to do it think of the drawing in 4 sections like an x/y grid that you used in geometry and algebra when you were in high school. You have your X+ and X- on the horizontal and Y+ and Y- in the vertical, so your four quadrants can be broken up into X+Y+, X+Y-, X-Y-, X-Y+. Put a basic sketch of what you want in each quadrant with no perspective, just as placement, then start making your grid. I suggest using a sort of numbering system for how many lines you want to have in the grid basically. So on each lets say make 10 increments at equal distance along each axis. Use measurements to make them even at first it will make altering them easier and more consistent, now as the move further away from the center have them increase slightly in distance each time at a small rate, the more drastic the rate the more skewed the perspective will be.
The next step is to start near the center an draw a line from each point towards the opposing vanishing points, curve them as necessary, the curves will become more drastic the farter away from the center you move.
Once you have your grid the trick is to treat each object in the way that the lines curve, so objects in the X-Y- section (lower left) will all curve up to the left and down to the right. Just match the edges to the lines of the grid you made for a guide. The most important thing is the fifth point, with is in the center, because not only will the objects curve but they will also grow as they approach the center. Try simple shapes first, then once you really understand it you can go complicated from there.
Making a grid and then drawing on top of it with tracing paper might be your best bet at first so that you can experiment without drawing a new one each time
Hopefully some of that made sense.