1 to the power of infinity, why is it indeterminate?

It isn’t: $\lim_{n\to\infty}1^n=1$, exactly as you suggest. However, if $f$ and $g$ are functions such that $\lim_{n\to\infty}f(n)=1$ and $\lim_{n\to\infty}g(n)=\infty$, it is not necessarily true that

$$\lim_{n\to\infty}f(n)^{g(n)}=1\;.\tag{1}$$

For example, $$\lim_{n\to\infty}\left(1+\frac1n\right)^n=e\approx2.718281828459045\;.$$

More generally,

$$\lim_{n\to\infty}\left(1+\frac1n\right)^{an}=e^a\;,$$

and as $a$ ranges over all real numbers, $e^a$ ranges over all positive real numbers. Finally,

$$\lim_{n\to\infty}\left(1+\frac1n\right)^{n^2}=\infty\;,$$

and

$$\lim_{n\to\infty}\left(1+\frac1n\right)^{\sqrt n}=0\;,$$

so a limit of the form $(1)$ always has to be evaluated on its own merits; the limits of $f$ and $g$ don’t by themselves determine its value.

The limit of $1^{\infty}$ exist:$$\lim_{n\to\infty}1^n$$ is not indeterminate. However$$\lim_{a\to 1^+,n\to\infty}a^n$$ is indeterminate..

There are many reasons. For example, let $1^\infty=1$. Taking logarithm, you have $\infty\cdot 0=0$. Similarly for other operations you will obtain some absurd.