Weird problem with ADXL345, Buzzer and Arduino
To expand from my comment:

This could easily be due to the buzzer itself: either physical vibration or EM noise. The fact that replacing the buzzer with an LED supports this as the LED gives no physical movement, and draws a nice steady amount of current. As the buzzer is on the same bread board as the LED, it is mechanically rigid to the sensor, while also being electrically close, and so it is tricky to say which out of physical or EM noise is more likely (my hunch would be physical movement which is easier to cause and easier to cure).

To confirm that it is the buzzer that is causing the issue I would recommend connecting the buzzer via some flying leads. This will mechanically isolate the buzzer from the sensor. When the test is repeated, if the sensor does not give the "funny" reading you are seeing now, the mechanical movement of the buzzer is the cause. The cure for this is then simple: mechanically isolate the buzzer from the sensor (put one or other or both on flying leads). If that is not something you can do for your application, you can either dampen the buzzer signal (using rubber mountings) or you could change your software to filter out the noise (which is often done using over-sampling and taking the mean of the data).

You have replied to my comment saying that the flying leads fix the issue, so you now have various options:
Mechanical: isolate the buzzer from the sensor in any way you like (flying leads, rubber mountings)
Software: sample the data and filter out fast changing values (averaging over a set period of time is the easiest)
Hardware: use a different sensor that samples at a slower rate (so can't pick up the motion from the buzzer). If you are using a different sensor with an analogue output, you could filter the output, with a simple capacitor.

As this appears to be mainly a software project, I would go for a basic software filter. The specifications of it will depend on your application; speed of response to change, fasted change you want etc. You can go a long way with filtering, depending on your application, using; averaging, rate of change, frequency matching, debouncing.