Think of heat death as the total opposite of the Big Crunch. In this case, gravity is not strong enough to overcome the expansion, so the universe just keeps on expanding exponentially. Galaxies drift apart like forlorn lovers, and the all-encompassing night between them becomes wider and wider.
The universe follows the same rules as any thermodynamic system, and they all end up the same way: with heat evenly distributed throughout. If we extrapolate that to our universe, we wind up with all of the matter evenly dispersed throughout—in a cold, dark, boring mist. Eventually, the stars will all wink out, one by one, and there won’t be enough energy left to ignite new ones.
Finally, the entire universe will go dark. Matter will still be there, but in particle form, and its motion will be totally random. The universe will be in a state of equilibrium, and these particles will bounce off of one another without exchanging energy. We’ll be left with just particles in a void.