In linear regression :
$$ y_i = \beta_1 + \beta_2x_i + \epsilon_i $$
I don't understand why Var$(y_i)= \sigma^2$
Because
\begin{align}
\text{Var}(Y_i) &= \text{Var}(\beta_1 + \beta_2 x_i + \epsilon_i) \\
&= \beta_2^2 \text{Var}(x_i) + \sigma^2
\end{align}
Best Answer
$x_i$ is one single non-random variable, so on itself it has a variance of 0, so the formula you wrote simplifies to just $\sigma^2$.
Normally $y_i$ is expressed as follows:
$$y_i \sim N(\beta_1 + \beta_2x_i, \;\sigma^2)$$
This way it should be evident how the variance of $y_i$ is determined. $\beta_1 + \beta_2x_i$ only contributes to the expected value of $y_i$.