Two Dimensional Tape of Turing Machine
The idea is very similar to the way we show that the rational numbers are countable.
Essentially, what you need to do is to find a (bijective) mapping of the 2D tape onto the 1D tape. Of course, you want this mapping to be computable, and it would be helpful if it is computable efficiently.

The illustrative way to think about this is in diagonals: the cell $(1,1)$ is mapped to cell $1$ in the 1D tape. Then the second diagonal, namely cells $(1,2)$ and $(2,1)$ are mapped to cells $2,3$ respectively. Continuing in the same manner, you can map the entire 2D tape to the 1D tape. Of course, any movement of the head needs to be translated to the correct position in the 1D tape, so a single movement in the 2D tape may actually take several operations to simulate in the 1D tape.

There are many explicit functions that compute such a mapping. See for example the Cantor pairing function.