To take a photo like this one you'll need a tripod and a camera.
I use the exposure lock functionality of my camera to make sure all photos are with same exposure. This will avoid brighter/darker areas of the combined photo. You can use manual mode of your camera instead of exposure lock.
Combining the photos is easily done in GIMP, which is free image editing software. Just add two of the photos to Layers window. Then right click the first one and select "Add Alpha Channel". Then use the Eraser tool from the Toolbox to erase the area where the object appears in the second layer only. Export the photo and repeat until all photos are combined.