I know that the energy of SHM is given by
$$E=\frac 12 kA^2$$
So which of these is more appropriate to say?
Energy is increases because amplitude is increased.
Amplitude is increased because energy is increased.
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It depends on the context of the question you're asked to solve using this formula. The formula you've written is that of the total energy of a simple harmonic oscillator, such as a block attached to a spring.
At the maximum displacement, there is no kinetic energy and the potential energy is hence given by $\frac 12 kA^2$. Note that from this, $E \propto A^2$.
Now, if we pull the block so that its amplitude increases, we are giving energy to the system. This means that the energy and the amplitude both increase.
Do note that we can't say one happened because of another, it really depends on the case you're looking at. (A parallel you can draw is, in $\vec F=m \vec a$, if you increase the force applied, the body's acceleration will increase, and if you note an increase in the body's acceleration, of course the force must be increasing.)
In most cases, however, we usually are able to easily change the amplitude manually, because of which the energy of the system changes. Changing the amplitude is the way in which we increase the energy. In that sense, the energy increase happened because we changed the amplitude.
There is no 'because' in an equation. $a=b$ does not imply $a \to b$ or $b \leftarrow a$.
If you increase the energy of an oscillator, you will increase its amplitude. If you increase its amplitude you will increase its energy.
This is brought out neatly in our use of Ohm's law. $V=IR$ is usually read and used as 'If you apply $V$ volts across $R$ ohms you get $I$ amps.' But sometimes (for example in a potential divider) you use it as 'There is a current of $I$ amps flowing through $R$ ohms so that gives $V$ volts".