Total subsets are $\,^6C_0 + \,^6C_1 + \,^6C_2 + \dots + \,^6C_6 = 2^6 = 64$
Another way to look at it is:
To make a subset from the original set, we have 2 choices for each element. Each element can either be in the subset, or not be in the subset.
Therefore, total possible ways $= \underbrace{2 \times 2 \times \ldots \times 2}_{6 \text{ times}} = 2^6 = 64$
