In general if a quantity $n$ is increased by say $7$ percent, then the outcome is the sum of the original $n$ and the increment, which is $0.07n$, and this yields $1.07n$. Essentially we are saying that with a $7$ percent increase the original quantity is now $107$ percent of the original.

In this particular question, the quantity $(2.08x)(0.08)$ you mentioned is the increment and this needs to be added to the balance of $2.08x$, which is equivalent to $(2.08x)(1.08)$. I hope this makes sense.

Aditi Gupta says

Why do we multiply 1.08 as 8% rate for the second year. Shouldn’t it be 2.08x * 0.08?

GMAT Quantum says

In general if a quantity $n$ is increased by say $7$ percent, then the outcome is the sum of the original $n$ and the increment, which is $0.07n$, and this yields $1.07n$. Essentially we are saying that with a $7$ percent increase the original quantity is now $107$ percent of the original.

In this particular question, the quantity $(2.08x)(0.08)$ you mentioned is the increment and this needs to be added to the balance of $2.08x$, which is equivalent to $(2.08x)(1.08)$. I hope this makes sense.