What is multivariable calculus?

For an Algebra 1 student, I would start by talking about tangent lines to curves in the plane. Without getting into detail, we can explain the intuitive approach of using secant lines to approach the tangent line.

Depending on the student's knowledge of coordinate graphing of lines, we can try to connect slopes of lines to rates of change.

The lift the discussion to three dimensions. Imagine a grid (coordinate system) laid out on the floor. (Start from a convenient corner so we can point out the third axis rising above the floor.)
Now imagine a surface (Maybe we can use the idea of a childhood hideout of blankets draped over furniture. If we live in a hilly area, we can point out the rises and dips outside a window.)
Now standing a some point determined by the grid, we can talk about the slopes in various directions. Maybe in this direction we see a slight uphill and in that direction a very steep uphill.
In different directions we may see different slopes.

With a sharp student we might be able to change the surface to some other measurement like temperature (on the 2-dimensional grid). In stead of a tangent line, we then can talk about rates of change in various directions.

If that works out, we can suggest a coordinate system for the room (3-dimensions) and talk about some forth measurement (I like temperature again) and its rate of change.