You can read off a simple proof from Section VII.2 of Bourbaki's General Topology IIPart 2. Roughly, the proof goes as follows. Let $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, then any continuous homomorphism $\mathbb{T}^m\to\mathbb{T}^n$ lifts to a continuous homomorphism $\mathbb{R}^m\to\mathbb R^n$ which then is linear and maps $\mathbb{Z}^m$ into $\mathbb{Z}^n$.
You can read off a simple proof from Section VII.2 of Bourbaki's General Topology II. Roughly, the proof goes as follows. Let $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, then any continuous homomorphism $\mathbb{T}^m\to\mathbb{T}^n$ lifts to a continuous homomorphism $\mathbb{R}^m\to\mathbb R^n$ which then is linear and maps $\mathbb{Z}^m$ into $\mathbb{Z}^n$.