Homoscedasticity is one of the Gauss Markov assumptions that are required for OLS to be the best linear unbiased estimator (BLUE).
The Gauss-Markov Theorem is telling us that the least squares estimator for the coefficients $\beta$ is unbiased and has minimum variance among all unbiased linear estimators, given that we fulfill all Gauss-Markov assumptions. You can find more information on the Gauss-Markov Theorem including the mathematical proof of the theorem here. Additionally, you can find a complete list of the OLS assumptions including explanations what happens in case they are violated here.
Briefly summarizing the information from the websites above, heteroscedasticity does not introduce a bias in the estimates of your coefficients. However, given heteroscedasticity, you are not able to properly estimate the variance-covariance matrix. Hence, the standard errors of the coefficients are wrong. This means that one cannot compute any t-statistics and p-values and consequently hypothesis testing is not possible. Overall, under heteroscedasticity OLS loses its efficiency and is not BLUE anymore.
However, heteroscedasticity is not the end of the world. Fortunately, correcting for heteroscedasticity is not difficult. The sandwich estimator allows you to estimate consistent standard errors for the coefficients. Nevertheless, computing the standard errors via the sandwich estimator comes at a cost. The estimator is not very efficient and standard errors might be very large. One way to gain back some of the efficiency is to cluster standard errors if possible.
You can find more detailed information on this subject on the websites I referred above.