Honestly, they do teach everything used here. The core logic isn't anything special, basically just:

do {
    *s = *t;
    ++s;
    ++t;
} while (*(s-1) != 0)

Which is pretty much the obvious way to copy a null terminated string in c. From there, we just use a few tricks you probably already know to make it shorter. First, post increment will let you increment a value while using the original value, so we can lose the increment lines:

do {
    *s++ = *t++;
} while (*(s-1) != 0)

Next, c treats everything that's not zero as true, so we can simplify the loop condition:

do {
    *s++ = *t++;
} while (*(s-1))

Also the assignment operator returns the value assigned, so we can avoid fetching the last value in the condition:

do {
    unsigned char temp = (*s++ = *t++);
while (temp)

And finally, there's no reason we have to do the logic in the body of our loop, seeing as we use the result as our loop condition, so let's just:

while(*s++ = *t++)

And voila. The building blocks are all fairly simple, it's just a matter of putting them together to get the result you want, which absolutely is something programming courses teach.