We have a grid of size $n \times m$. An ant starts at $(0 ; 0)$ and wants to reach the cell $(n ; m)$, but it can only move upwards or to the right. How many paths can it take?
When the ant arrives at $(n ; m)$, it has performed $n$ moves to the right and $m$ moves upwards. Consider the sequence of our moves to the right, for example. We then need to find how to position our $m$ moves upwards among all our total moves, which gives us: $\binom{n+m}{m}$ possible paths. But we can also reason with $n$, and we have: $\binom{n+m}{n}$, which is indeed equal to $\binom{n+m}{m}$.