A n-dimensional variable is just a variable with n independent single variables. If it is a n-dimensional random variable, it's just a function of n individual random variables.
Its exactly the same as a non-random function of many variables like f(x,y,z) = x^2 + y + z but these variables are now random variables.
In probability when you have n-dimensional random variables, you have what is called a joint distribution and this joint distribution is a distribution that tells you P(A = a, B = b, C = c, etc) at those particular points so you can think of this new PDF as being a probability for all different combinations of possibilities for all of the random variables.
Before in univariate situations you only have P(X = x) or P(Y = y), but now you have more degrees of freedom which means that you have to now introduce more ways to sum or integrate if you want to get a probability over that region.
It has the same interpretation probability wise as the univariate but since you have multiple variables now, you also have co-relationships between the variables which gives rise to covariance, correlation and other things.
In terms of the random variables themselves, the joint distribution is defined across all permutations of each of the random variables, but you can still calculate expectations like X + Y or X - Y or X^2 + Y^2 just like you do in the normal univariate case by using E[g(X,Y)] instead of E[g(X)].
But that's all it really is.