Why is the nature of graph of utility function different from indifference curve? I am new to Economics, but I have this doubt. The indifference curve and utility function both have the same equation, so their graph must also be similar, which is true I guess. Then why is it that the nature of graphs are different? I mean why is it that if the indifference curve is convex then utility function is quasi concave?  How did we arrive at this conclusion?
You seem to be confusing some things.


  The indifference curve and utility function both have the same
  equation


The utility function has a formula, not an equation.

$U(x_1,x_2)$ is a utility function.

The points $(x_1,x_2)$ for which $U(x_1,x_2) = 13$ form an indifference curve. (13 was picked as a random number)

The graph of a utility function has one more dimension than the graph of an indifference curve: in addition to the space of goods, this graph would also include the utility value, so its points would be something like $(x_1,x_2,u)$. Chances are you have not seen a graph of any such function, as 3D drawings are complicated. This is why maps of 2D indifference curves are used to get a feeling for the shapes of utility functions.