The site you got the circuit and code from tells you what it is doing.
From the description, you can find out what the output looks like. It says it is using pulse width modulation (PWM) to generate the audio.
Wikipedia has a good explanation of PWM, and how it is used to make audio.
This image from the wikipedia page show pretty much what you can expect as output from your noise maker.
The red line is what your speaker "sees." The blue line is what your circuit generates.
Simply put, your program makes pulses of various widths on the two output pins. Wider pulses cause the speaker to move more than narrower pulses. The pulses have to occur faster than the speaker can respond to them. The inertia of the speaker smooths the pulses into something that resembles normal audio.
What is limiting the volume of the output is the current from the output pins. That is normally measured in milliamperes. Assume you can get 20mA out of your pins. At 7 volts and 20mA, you will get maybe 0.14 watts. Assuming you don't kill your processor doing this.
That's not much power. Compare it to what you expect from your stereo, and how many watts of power it can output.
To get more power, the simplest thing to to is to clean up the PWM signal and send it to a regular amplifier.
You only need one of the outputs pins. Your gadget uses two in attempt to get more power, but it really can't do much.
So, do this:
simulate this circuit – Schematic created using CircuitLab
Then connect the output to any standard audio amplifier. Buy and build a kit if you like, or connect it to the line in of a stereo.
That is a low pass filter. It filters out everything above 4kHz. I chose 4KHz because the project says to use a file with an 8kHz sampling rate. A file sampled at 8kHz can't contain any audio above 4kHz. Look up Shannon and Nyquist (in relation to sampling rates.)
The pulse rate of the PWM will have to be much higher than 8kHz, so using a 4 kHz filter doesn't lose any of your audio, but should smooth the square edges into a nice smooth wave.
There are small amplifiers that you could power from 3.3V. There are also many other amplifiers that run on higher voltages. There are cheap and common circuits (and kits) to build an amplifier from an LM386. The LM386 needs 5V or more, though. It is commonly operated on a 9V battery.
You could also look up H-bridges. If you drive an H-bridge with you speaker in the middle (the cross bar of the H) using your two processor outputs on the H-bridge inputs, then it should be quite loud. This could be tricky, though. It will be operating at the pulse rate, and amplifying the pulses rather than the smooth wave. That will cause a lot of electrical "noise" and probably make your speaker and the bridge get hot.