First, you should know that the chromatic number of the graph is the minimum number of color required to color a graph so that no two colors are at the adjacent node.
Now if you read the above statement carefully, then you will understand that you can not assign the same color to the node which has a direct edge, and if they do not have a direct edge then you can definitely assign them the same color.
So, if you consider the above two statement very carefully, then you will come up to RHS of what you are asking that for a graph, the maximum color we need is the ( (max degree of the graph) + 1 ).
But don't always need that much of color to color the graph, sometimes we need very very less than that. Check these simple example:
Here the max degree is 3, hence at max you need (3+1) color, but you can easily color this graph by using 2 color only.
Check this one, it has max degree 2, hence it may need max 3 color, but you can easily color this by using 2 color.
Chromatic Number <= (max degree of graph) + 1