You've actually already put together all the right pieces in your question. The cdf is
- along one axis, say $x$, a staircase (similar to stacking books atop each other on a table)
- along the other axis, say $y$, continuous
So, imagine a staircase where each step has its own continuous curve, and each successive step's curve is greater than or equal to the last step's curve at every point.
Or, looking at it from the perpendicular direction, each part of a continuous curve has a corresponding staircase, each staircase greater than or equal to the last at every point.
As you say, the cdf goes to 1 as $x$ and $y$ go to $(\infty, \infty)$.
This plot of a multivariate CDF is for a Binomial($1/3$,$2$) variable $X$ and an independent Beta$(3,4)$ variable $Y$. Because the variables are independent this CDF is the product of the individual CDFs.