What's the precise meaning of the expression "induced by" in mathematics?

First, "induce" is a perfectly cromulent English word. The second definition that Google gives is relevant here:

bring about or give rise to.

In basic vernacular English, it is reasonable to say that "$A$ induces $B$" when $A$ causes $B$, though I think that there is a connotation of indirectness (i.e. there might not be that $A$ directly causes $B$, but $A$ creates the conditions for $B$). In mathematics, this is the definition that is generally meant. When we say that "$A$ induces $B$," we typically mean that $A$ gives rise to $B$, typically in some canonical manner.

For example (in an area with which I am more familiar), we often say that a "metric induces a topology". What this means is the following: if $(X,d)$ is a metric space, then the open balls, i.e. the collection $$ \mathscr{B} := \{ B(x,r) : x\in X, r> 0 \}, $$ where $B(x,r) := \{ y \in X : d(x,y) < r \}$, forms a basis for a topology on $X$. The topology generated by this basis is the topology induced by the metric. That is, the metric gives rise to this topology.

After a bit of Googling, a "planar subdivision induced by a set of $n$ line segments" seems to make sense in a similar way. Near as I can tell, a planar subdivision is a partition of the plane, i.e. a division of the plane into a collection of mutually disjoint sets whose union is the plane. A partition has more structure than just a collection of line segments, but a collection of line segments can give rise to a partition in a canonical manner. It is therefore appropriate to say that such a partition is induced by a collection of line segments.

In the specific case, the $n$ line segments can be extended uniquely to lines, that give a planar subdivision (Edit: see comments below). In general, as mentioned in the comments, there is no literal interpretation that always works. Sometimes we have a "smaller" thing that can be extended uniquely (as in this case), sometimes we have a " bigger" thing that can be reduced uniquely (as for the induced or relative topology on a subset of a topological space), and sometimes neither of the two. A good use of the word requires that it is clear from the context what is meant. The idea is always to adapt /modify something given in a canonical / unique way to get what we need.