Handedness is not constant over the animal kingdom.

Kangaroos and wallabies tend to be left handed, though wallabies seem to be right handed for strength tasks. In dogs, horses, and cats females have been shown to be left handed, while males are right handed. All of these are tendencies and not at all strict, so specific inviduals may be left or right handed with no regard to their sex or species, but the trend is there.

The level of handedness that humans have is really white extreme compared to our closest cousins. Other primates are far less handed and one of the things that drives this may be tool use and associated teaching. If you are teaching someone how to do something and they have opposite handedness to you it is harder to teach, and also shared tools are easier to manage if they are not in two different versions.

The causes seem to be a mix of genetics, developmental cues, and maybe brain structure, though the exact amounts of each and whether there are other factors are unclear.