The easiest method of approach i can see for this problem is as follows:
For the functionality of the navigation (client side) it would be easiest if you used a jquery carousel plugin to implement your bottom navigation set.
For the changing of the videos there are several options, but here are 2 of the most likely one’s you could implement:
For each navigation item link to a new page for each video (have a seperate .html, .aspx, .php page etc for each video)
If you want to just use one page and make it dynamically the video load in the page, i’d recommend using a Query string for each video link (linking to the same page), then have the back-end fetch this value and replace the id (see values in red) within your youtube embed tag…e.g: