Multivariate Gaussian Distributions

I am interested in understanding Multivariate Gaussian Distributions. I am a (young) Computer Scientist looking to implement the EM algorithm for mixture models.

I understand univariate normal distributions and the rationale behind them. I've looked up multivariate normal distributions but can't wrap my head around all the information on the internet. Could someone spare their time to show me the rationale behind our formalization of multivariate distributions?