small floresent tube removal from a wooden dislay unit
Reverse engineer the tube

Go to the store and find the replacement tube. Look at the ends.

If they are single nubs (probably not), you slide the tube linearly to compress a socket at one end - anyone's guess which.   

If you see two pins on each end, the vast majority of those involve rotating the tube 90 degrees (or nearly so), then it slides/wiggles straight out.  A small number involve the tube snapping straight in.  

When choosing a new tube, look closely at CRI (color rendering index).  Older fluorescents were positively pallid.  New fluorescents must be 80 CRI or better, which is quite good.  It's like steam locomotives and carburetors, they got really excellent just as they were going obsolete. 

Hulk! Smash!  Then go with 12V LED strips

The other option is to remove the tube ungracefully, mindful that fluorescent tubes contain a very small amount of mercury.  

Then obtain common "LED strips" in either monochrome in any of several color temperatures (including adjustable), or RGB or RGBW color... Obtain a dimmer/blender/controller if you want that, and a 12V DC power supply that plugs into the wall.   Many outfits on Amazon and eBay sell this stuff a-la-carte, or in complete kits for about $20 for the works, including 16 feet of LED strip.  Better vendors charge more (often, usuriously more) but have better control of CRI.   

You should brace yourself for the fact that LEDs never fail, so take care to choose the right stuff the first time, as there won't be a second.   Power supplies and controllers do fail, but are cheap and easy to replace.