Why does it mean that the process/signal is not stationary when its variance varied with time?
From the Wiki page - A stationary random process is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance, if they are present, also do not change over time.

Additional References are: Link1 and Link2.

The implication from the definition is that the mean and variance of random process do not vary over time. Since by your definition of the variance - it varies over time, then your process is non-stationary.

A couple of notes:


Just because the mean and variance don't vary over time does not imply that the process is stationary. The definition of a stationary random process has more implications.
A random process where the mean and auto-covariance are invariant to time shifts is often called a "Wide Sense Stationary" process. There are no conditions on the higher order moments (skewness, kurtosis, etc). This is a less stringent version of stationarity that is often more applicable for real-world signals.