Here's how I think about it. Don't worry too much about the equation.
The thin film seems to be surrounded by air like a sandwich: air - film - air. The film has a higher refractive index than the air.
Light will bounce off the first interface of the material (going air to film) and phase shift because the film is more 'dense' and acts as a fixed boundary.
However, the light that passes through and hits the second interface does NOT phase shift. Film to air is going to a less 'dense' medium, so it is a free boundary.
This means that the waves are exactly half a wavelength or 180 degrees out of phase (one wave phase shifted and the other didn't). How can you get them back in phase for constructive interference? By making one travel longer in such a way that they get back together.
The second wave traveled a longer way. It passed through the thin film twice; once entering, once leaving. It has a path length difference of 2 * thickness. You can manipulate this second wave by changing the film thickness so that the extra distance traveled equals half a wavelength (the phase shift from before). This will cause the waves to be in sync.
Path length difference (2*thickness) must equal lambda/2.
2*thickness = lambda/2
thickness = lambda/4
This shows that the film thickness has to be a quarter wavelength. Two times a quarter wavelength will give us a half wavelength, which is exactly what we need for re-syncing.
But wait. Wavelength changes when going to a new material! This new wavelength is what you need to work with and plug into lambda/4 eventually.
v = c/n; the speed of light will slow down by n (refractive index) to a new and slower velocity v. Since velocity drops, wavelength drops by the same factor n (frequency is pre-determined and never changes once light leaves its source). Note that the speeds of light in air and vacuum are practically the same, so you are allowed to do this.
The lambda we need, therefore, needs to be divided by n = 1.5 to find the new wavelength in the film. 480nm / 1.5 = 320 nm.
Now, lambda/4 = 320nm / 4 = 80 nm. . This is the thickness of the film needed, taking into account the changing wavelength of light in the film, in order to cause a path length difference of lambda/2 to get the two waves to sync constructively. This is the MINIMUM thickness. It could be thicker to get the same results, but a quarter wavelength is as small as it gets to achieve success.
The 'm' in the equation is a whole number. If you set it to zero, you get the answer seen here. (m+0.5) pretty much tells you that if you set m=0, you get a 2*thickness = 0.5lambda relationship, which is exactly what we had before just thinking about it conceptually. By the way, the n on the left side can be divided over to the right and put under lambda and this will give you the wavelength in the material, which is also in line with what's above.
Long post, it's late and I hope it helps.
