Suppose you have a random variable $Y$ that for simplicity you assume is normally distributed. So
$$Y \sim N(\mu, \sigma^2) = \mu + \epsilon \quad \text{ where } \epsilon \sim N(0, \sigma^2)\,. $$
If you knew $\mu$, then an unbiased estimator of $\sigma^2$ will be
$$S_1^{2}=\dfrac{1}{n}\sum_{i=1}^{n}\left(y_{i}- \mu\right)^{2}\,.$$
If $\mu$ is not known, then you can estimate $\mu$ with $\bar{y}$. Now an unbiased estimator of the variance $\sigma^2$, as you mention would be
$$S_2^{2}=\dfrac{1}{n-1}\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}$$
The $n$ in the denominator changed to $n-1$ because some effort now has gone into estimating $\mu$. This is connected to the concept of degrees of freedom. The link has some nice discussions on this.
Now suppose you don't know $\mu$, but you know that it is of the form $\mu_X = \beta_0 + \beta_1X_1 + \dots + \beta_p X_p$ where the $X$s are given to you and $\beta$s are unknown. This is the linear regression model.
$$\mu_X = E(Y \mid X) = \beta_0 + \beta_1 X_1 + \dots + \beta_p X_p\,.$$
Since $\beta$s are unknown you estimate them using ordinary least squares, and calculate an estimator for $\mu_X$
$$\hat{y}_i = \hat{\mu}_X = \hat{\beta}_0 + \hat{\beta}_1X_ + \dots + \hat{\beta}_pX_p,. $$
In estimating $\mu_X$, we had to estimate all of $\beta_0, \beta_1, \dots,\beta_p$, which are $p+1$ parameters. So now an unbiased estimator is
$$S_3^{2}=\dfrac{1}{n-p-1}\sum_{i=1}^{n}\left(y_{i}-\hat{y}_i\right)^{2}\,.$$
Thus, the change in the denominator comes from the fact that each time we are using a different estimator for the mean $\mu$.
