In your mind you only know 10 different digits. 0 to 9. Internally in your brain, this is certainly encoded differently than in a computer.
A computer uses bits to encode numbers, but that is not important. That's just the way engineers chose to encode stuff, but you should ignore that. You can think of it as a 32 bit computer has a unique representation of more than 4 billion different values, while us humans have a unique representation for 10 different values.
Whenever we must comprehend a larger number, we use a system. The leftmost number is the most important. It is 10 times more important than the next.
A computer able to differentiate between four billion different values, will similarly have to make the leftmost value, in a set of values, be four billion times as important as the next value in that set. Actually a computer does not care at all. It does not assign "importance" to numbers. Programmers must make special code to take care of that.
Whenever a value becomes greater than the number of unique symbols, 9 in a humans mind, you add one to the number to the left.
In this case, the number still fits within a single "slot"
5+5=10. This situation is called an overflow.
So humans always deal with the problem of not having enough unique symbols. Unless the computer has a system to deal with this, it would simply write 0, forgetting that there was a number extra. Luckily, computers have an "overflow flag" that is raised in this case.
987+321 is more difficult.
You may have learned a method in school. An algorithm. The algorithm is quite simple. Start by adding the two leftmost symbols.
7+1=8, we now have ...8 as the result so far
Then you move to the next slot and perform the same addition.
8+2=10, the overflow flag is raised. We now have ...08, plus overflow.
Since we had an overflow, it means that we have to add 1 to the next number.
9+3=12, and then we add one due to overflow. ...308, and we had another overflow.
There are no more numbers to add, so we simply create a slot and inser 1 because the overflow flag was raised.
A computer does it exactly the same way, except it has 2^32 or even better 2^64 different symbols, instead of only 10 like humans.
On a hardware level, the computer works on single bits using exactly the same method. Luckily, that is abstracted away for programmers. Bits is only two digits, because that is easy to represent in a power line. Either the light is on, or it is off.
Finally, a computer could display any number as a simple sequence of characters. That's what computers are best at. The algorithm for converting between a sequence of characters, and an internal representation is quite complex.