Hello)
This is not an easy question and I hope everyone not to have it on the exam)
Actually this problem is connected with wide and very interesting field of mathematics - graph theory(trees, cycles, 7 bridges of Konigsberg etc.). Probably you have heard about it. If no and you are interested - see here.
http://en.wikipedia.org/wiki/Graph_theory
However, I will try to explain it to you without any extra knowledge.

Obviously, the shortest way is to go to the right and up. I think it is easy to understand. So, the total lenght of the trip is 9 steps. Since you go from the LeftDown to RightUp corner, then you always will go through one of the "key" points, as I call them. See, i circle them in red in the picture.

Now let calculate, how many routes are there to the point 1. Obviously, there is 1 such route. I show it in yellow.

Now let calculate how many routes are there to the point 2. There are 4 such routes. You could easily understand these looking at the picture. I show these routes in yellow, green, blue and red.

How many routes are there from the origin to the 3 point? 6! See the picture.

How many routes to the 4 point? There are 4. Everything is on the picture.

And finally there is only one route to the 5 point. I hope you understand this easily.

And now let do the second part of work. Let calculate, how many options you have going from each of these 5 points to the final point - RightUp corner!
For the 1 point there is only 1 such route - going to the right.
For the 2 point there are 5 routes. See the picture, please.

From the 3 and 4 points there are 4 routes, They are quite similar, so I will show the picture only for one case.

Finally, from the point 5 there are 5 routes to the end, it is like the point 2, so I do not add the picture.

Now we are close to the answer!
What we need is to calculate how many routes, going through each of the 5 "key" points, is there. As you guess we should just multiply the number of routes from the first and second parts.
So, there are 1*1=1 routes going through point 1.
4*5=20 routes going through point 2
6*4=24 routes through point 3
4*4=16 routes through point 4
1*5=5 routes through point 5

The final answer is 1+20+24+16+5=66!

Obviously, this is not the best or shortest way to solve this, but I hope it will help to realise better the essence of the
problem. I sorry for the "quality" of pictures, but I have not much time to draw them)
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Thanks fluke & bagrettin.
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Thanks Karishma for this awesome explanation! I was scratching my head on a GMAT problem for the very first time in my GMAT prep. I can use this strategy in many problems Awesome!