What does flushing the buffer mean?

Consider writing to a file. This is an expensive operation. If in your code you write one byte at a time, then each write of a byte is going to be very costly. So a common way to improve performance is to store the data that you are writing in a temporary buffer. Only when there is a lot of data is the buffer written to the file. By postponing the writes, and writing a large block in one go, performance is improved.

With this in mind, flushing the buffer is the act of transferring the data from the buffer to the file.

Does this clear the buffer by deleting everything in it or does it clear the buffer by outputting everything in it?

The latter.

You've quoted the answer:

Output buffers can be explicitly flushed to force the buffer to be written.

That is, you may need to "flush" the output to cause it to be written to the underlying stream (which may be a file, or in the examples listed, a terminal).

Generally, stdout/cout is line-buffered: the output doesn't get sent to the OS until you write a newline or explicitly flush the buffer. The advantage is that something like std::cout << "Mouse moved (" << p.x << ", " << p.y << ")" << endl causes only one write to the underlying "file" instead of six, which is much better for performance. The disadvantage is that a code like:

for (int i = 0; i < 5; i++) {
    std::cout << ".";
    sleep(1); // or something similar
}

std::cout << "\n";

will output ..... at once (for exact sleep implementation, see this question). In such cases, you will want an additional << std::flush to ensure that the output gets displayed.

Reading cin flushes cout so you don't need an explicit flush to do this:

std::string colour;
std::cout << "Enter your favourite colour: ";
std::cin >> colour;