Making a bee-line with Fibonacci numbers

Here is a picture of a bee starting at the end of some cells in its hive. It can start at either cell 1 or cell 2 and moves only to the right (that is, only to a cell with a higher number in it). There is only one path to cell 1, but two ways to reach cell 2: directly or via cell 1. For cell 3, it can go 123, 13, or 23, that is, there are three different paths. How many paths are there from the start to cell number n?

The answer is again the Fibonacci numbers. Can you explain why?