One perspective (heh) involves the following relation among position vectors: $$\vec{r}_{A\rightarrow C} = \vec{r}_{A\rightarrow B} + \vec{r}_{B\rightarrow C}.$$ These position vectors can be for anything; object $A$ could be a house, object $B$ an ant, and object $C$ a leaf on the river. Here's a diagram to help:
So if you want to know the position of object $C$ relative to object $A$ (the bold dark arrow), you just have to know the position of some other object $B$ relative to those others.
To answer your question, you can apply this same idea to the solar system: $$\vec{r}_{\mathrm{Sun} \rightarrow X} = \vec{r}_{\mathrm{Sun} \rightarrow \mathrm{Earth}} + \vec{r}_{\mathrm{Earth} \rightarrow X}.$$ Or in pictures:
The position of planet $X$ relative to the sun (bold dark arrow, which is what we want) can be found if we know Earth's position relative to the Sun and planet $X$'s position relative to Earth. In this way, measurements of a planet's position as measured from here on Earth can be used to get a map of the solar system..
There is the added complication of knowing distance to planets and coming up with a convenient coordinate system in order to actually come up with values for these position vectors. Others may have better information on that.