In order to understand heat death, you need to know a bit about entropy.
If you look up entropy, you'll often see the definition "The total disorder of a system". This is kinda true.
A better way to understand entropy is to understand macrostates and microstates. Imagine if you had 100 coins. Two questions you could ask about your system of coins are "how many total coins are in the state 'heads', and what state are the individual coins in?". So the answer to your first question may be "There are 48 heads (and 52 tails)", and the answer to your second question may be "The first coin is a heads, the second is a heads, the third is a tails" etc.
The state of the individual coins is the "microstate". The state of the entire system is the microstate. So our macrostate is "48 heads". I'm obviously not going to list all 100 microstates lol.
Note that every microstate has the exact same probability. The 1st coin has a 50% chance of moving to heads after the next toss, and 50% chance of tails in the next toss (in general, systems wont always have the exact same chance of every state. This example just happens to).
Now let's see how many different microstates are available to this system. There are 2 choices 100 times, so the rules of probability would state that there are 2^100 states, each of which has the probability 1/2^100.
Let's look at an individual microstate. Consider the state where every single coin is on heads. There's only one way to arrange the coins so that each coin is on heads (I'm assuming the coins are labeled so you can't switch coin 1 and coin 2 and say "Tada! I have a new state").
Now consider the state "Every coin is on heads except coin number 50". There's exactly one way to do this, and that's to have every coin but 50 on heads, and 50 be on tails.
Now consider the state "Every coin is on heads except coin number 70". Again, there's exactly one way to do this. Every coin but 70 will be on heads, and 70 will be on tails.
But let's look at the macrostate of these two examples. The microstate of the first example "Every coin is on heads except coin number 50" has the macrostate "99 heads". The microstate corresponding to the second example "Every coin is on heads except coin 70" has the macrostate "99 heads", which is exactly the same as the first example. As a matter of fact, there are 100 ways to arrange the coins so that 99 are on heads (i.e. exactly one is on tails. You can make the first coin tails, or the second, or the third...etc).
So there's 1 microstate state corresponding to the macrostate "100 heads". But, as we've just shown, there are 100 microstates corresponding tot he macrostate "99 heads". So there are 100 times more states corresponding to "99 heads" than "100 heads". So if you randomly toss 100 coins, you are 100 times more likely to get "99 heads" than "100 heads".
Now consider the macrostate corresponding to "98 heads". If my math is right, there are 9900 ways to arrange 100 coins this way. So you are 9900 times more likely to get 98 heads than 100 heads, and 99 times more likely to get "98 heads" than "99 heads". As you can see, the more heads you add, the higher the probabilities become. Once you hit 50, you reach your maximum (3.068518756254967*10^93, which is 3 billion trillion trillion trillion trillion trillion trillion trillion times more likely than 100 heads. a 3 with 90 zeroes following). Once you reach less than 50 heads, the probabilities go back down (because, for example, 1 head is the same thing as 99 heads because 1 heads means 99 tails).
Most people would consider 100 coins on heads is an "organized set". The same would be true of "100 tails". It's an orderly way to arrange your coins. As you can see, the orderly ways to arrange things are MUCH less likely than the disorganized states.
Entropy is a number that measures how the amount of microstates corresponding to to a given macrostate. As you've seen, a macrostate with more corresponding microstates is much more likely. Typically, the disorganized states are WAY more likely.
Think about how easily your room gets messy. There are only a few arrangements of your room that correspond to the macrostate "clean". There's a limited ways to set up your room so that it's a clean room. But there are tons of ways a room can be messy. So it's easier to get your room messy than to get it clean. That's why your room gets messy easily without you even trying, but you usually have to work specifically to get it clean.
So what does this have to do with science or heat death? In physics, the macrostates correspond to quantities such as pressure, volume, and temperature. The whole purpose of the example above was to show you how much more likely high entropy states are than low entropy states. Even with just 100 coins, the highest entropy state was waaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaay more likely than the lowest entropy state.
Suppose you put the coins on a table that shakes in a way that causes the coin to flip. Put the coins in any state you like. Put them in the all heads state if you want. No matter how you put it, each flip is MUCH more likely to push the coins into a higher entropy state (i.e. a state with closer to 50 coins) than they are to be pushed into a low entropy states. And this is with just 100 coins. If there were a thousand coins the probabilties of high entropy states gets WAAAAAY higher. Unfathomably higher. Now consider the fact that macroscopic systems typically consist of MUCH more particles. One mole of a substance is on the order fo 10^23. So imagine how much more probable the high entropy states are here, in just one mole. The probabilities are so incredibly extreme that we make a law out of it. It's the second law of thermodynamics, and it states that the entropy of a system will always decrease. As you can see, there is a chance that the system could go into a lower entropy state, the the probability is so incredibly ridiculously unfathomably low that you can rest assured that we'll never see it happen in this universe (well I guess given an infinite amount of time it would eventually happen, but it would take an extraordinary amount of time).
Sorry for the huge explanation, but if you're wandering around the wiki pages and such, you need some sort of idea about entropy in order to understand heat death.
So we can rest assured that every system in this universe is always evolving towards a high entropy state. But why is entropy important? What does it do?
Well, there is a maximum amount of entropy in any system. So what happens if the universe reaches its maximum entropy? Since every process increases entropy, and entropy is at its max, then no processes can take place. Nothing can be done in a universe with max entropy. No stars will burn, no chemical reactions, no nothing. The universe is essentially dead. Also note that entropy is related to temperature. Notice how we have really hot suns living in a really cold vacuum? Well, in a max entropy, the temperature of everything evens out. Every part of the universe will be the same temperature. There will be no variations from place to place. No hot suns in the middle of a cold vacuum (like I said, stars can't exist at this point anyways). Every part of the universe will be the same temperature (and that temperature will be very close to absolute zero. Heat death is actually very cold. We call it "heat" death because of how heat is closely associated with entropy).
Everything we do in our lives depend on entropy. The efficiency of power plants depend on entropy. Power plants work by taking advantage of lower entropy systems. Our bodies can extract energy from food and use it because of the lower entropy of foods. Every energy source is derived from lower entropy things. So even if we were androids who didn't need stars or planets or food, we still couldn't survive heat death because there would be no way to have a power source. There's no way to extract energy from a higher max entropy system.
Like I said, high entropy states are so much more probable that heat death is inevitable if the universe survives long enough. There's no avoiding it. The universe will certainly keep going towards that state until it reaches it.